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Common Core Math 3 Quadratic & Polynomial Modeling
APEX %HIGH%SCHOOL%1501 $LAURA $DUNCAN$ROAD$
APEX , $NC $27502 $
Common%Core%Math%3%%Quadratics%&%Polynomials%
Day% Date% Homework%
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Common%Core%Math%3%–%Quadratics%and%Polynomials%
Topics%in%this%unit:%! Quadratic%Functions%
o Standard'and'Vertex'forms'
o Graphing''
o Solve'by'factoring,'taking'the'square'root,'quadratic'Formula,'graphing,'and'
completing'the'square'
o Derive'quadratic'formula'by'completing'the'square'
o Complex'numbers'and'operations'with'complex'numbers'
o Quadratic'modeling'
o Definition'of'quadratic'with'focus'and'directrix'
o Derive'quadratic'equation'from'focus'and'directrix'
o Sum'and'product'of'roots'(Honors)'
! Polynomial%Functions%o Definition'of'a'polynomial'function%o Standard'form,'and'classify'by'degree'and'#'of'terms%o Dividing'with'long'division'and'synthetic'division%o End'behavior'and'sketching'graphs%o Zeros,'roots,'solutions,'xHintercepts%o Remainder'and'Factor'Theorems%o Rational'Root'Theorem'(Honors)%
Students will be able to . . . " Solve'quadratic'equations'with'real'coefficients'that'have'complex'solutions.!" Complete'the'square'in'a'quadratic'expression'to'reveal'the'maximum'or'minimum'value'of'
the'function'it'defines.'!" Solve'Quadratics'by'factoring,'graphing,'inspection,'square'roots,'completing'the'square,'
and'quadratic'formula.'!" Derive'the'quadratic'formula'using'completing'the'square.'!" Add,'subtract,'and'multiply'complex'numbers'!" Derive'the'equation'of'a'parabola'given'a'focus'and'directrix.'!" Write'a'quadratic'equation'from'its'roots.'(Honors)!" Create'equations'in'two'variables'to'represent'relationships'between'quantities.!" Understand'the'relationship'between'zeros'and'factors'of'polynomials.'!" Divide'polynomials'using'long'and'synthetic'division.!" Graph'polynomial'functions,'identifying'zeros'when'suitable'factorizations'are'available,'and'
showing'end'behavior.''
" Use'the'remainder'theorem'to'factor'polynomials'of'third'degree'(or'higher'–'Honors).'
" Use'the'rational'root'theorem'to'factor'polynomials'(Honors)'
VOCABULARY
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a,b,&c "
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! 5,-"!"#$%#&'()*,%/"7#"#$""
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2- 4ac
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y = a(x - h)2
+ k
Quadratics - Day 0 Worksheet
Multiplying Polynomials Simplify and write your final answer in standard form when possible. You must show your work to receive credit. 1. (n + 6)(n + 7) 2. (y + 5)(y – 8) 3. (k + 12)(3k – 2) 4. (a – 3)(a² - 8a + 5) 5. (6x² - 5x + 2)(3x² + 2x +4) 6. (y + 4)² 7. (4 – 6h)² 8. (x + 7)(x – 7)
Factoring Polynomials Factor the following by grouping or GCF. You must show your work to receive credit. 1. 24x + 48y 2. 223015 dccd + 3. ababba 7284 22 ++ 4. 12y² + 9y + 8y + 6 5. 18x² - 30x – 3x + 5
Factoring Trinomials Factor the following using the factoring by grouping or box method. If not factorable, write “Prime.” You must show your work to receive credit. 1. c² + 12c + 35 2. -72 + 6w + w² 3. d² - 7d + 10 4. x² - 13xy + 36y² 5 g² - 4g – 45 6. 2b² + 10b + 12 7. 6a² - 17a + 12 8. 15n² - n - 28 9. 14k² - 9k - 18 10. 10x² + 21x - 10
Factoring the Difference of Two Perfect Squares Factor the following using the factoring by grouping, box method, or using the shortcut for difference of squares. If not factorable, write “Prime.” You must show your work to receive credit. 1. x² - 81 2. 6 – 54x² 3. 4x³ - 100x 4. 16n² - 25 5. 16a² - 9b² !
QUADRATIC REVIEW
Quadratic Function:
€
f (x) = ax 2 + bx + c where
€
a,b,&c are real numbers and
€
a ≠ 0
! standard form
€
y = ax 2 + bx + c " quadratic term: ax2, linear term: bx , constant term: c
! vertex form
€
y = a(x − h)2 + k where
€
a,h,&k are real numbers and
€
a ≠ 0
! the graph is called a parabola " If the “a” is positive (+), the parabola opens up. " If the “a” is negative (–), the parabola opens down
! to graph a quadratic
" start with the parent graph (
€
y = x 2), apply transformations and use the pattern
" make a T-chart " use your calculator
! axis of symmetry
" in standard form
€
x = −b2a
" in vertex form x=h
! vertex: the high point or low point of the graph
" in standard form
€
x = −b2a is the x coord. of the vertex
" in vertex form the vertex is (h,k) " the x coord. of the vertex is always the x value halfway between the x-
intercepts " to find the y coord. plug in the value of the x coord. and solve for y " use the calculator (2nd Trace > minimum or maximum)
! x-intercepts: points where the graph crosses the x-axis
" the real values of x that make y=0 " possible number of x-intercepts: 0, 1, 2
! zeros, solutions, roots: the values of x that make y=0
" may or may not be real " real zeros, solutions, roots are x-intercepts
! quadratic equation –
ax2
+ bx + c = 0 , to solve:
" use the calculator (2nd
Trace > zero)
" when there is no linear term, set y = 0 and solve for x (take the ± square
root)
" by factoring, set each factor = 0 and solve for x
" standard form: use the quadratic formula
x =-b ± b
2- 4ac
2a
! discriminant is the value of
b2
- 4ac
"
b2
- 4ac = 0 - one real rational double root; vertex of parabola lies
on the x-axis
"
b2
- 4ac > 0 and a perfect square - two real rational roots;
parabola intersects x-axis twice
" - two real irrational
roots; parabola intersects x-axis twice
"
b2
- 4ac < 0 - no real roots, two complex conjugate/imaginary roots;
parabola does not intersect the x-axis
! Using the calculator:
Enter your quadratic into Y=
(Be sure to use X as your independent variable)
To Find a Vertex (Maximum/Minimum):
1. Enter equation in Y =
2. Use CALC menu (2nd
TRACE)
Choose #3: minimum or #4: maximum
3. Move curser left/right until it is to the left of the vertex (close to point). Press ENTER
4. Move curser left/right until it is to the right of the vertex (close to point). Press ENTER
5. Press ENTER to reveal vertex (max/min)
To Find Zeros/Roots/X-Intercepts:
1. Enter equation in Y =
2. Use CALC menu (2nd
TRACE) Choose #2: zero
3. Move curser left/right until it is to the left of the zero (close to point). Press ENTER
4. Move curser left/right until it is to the right of the zero (close to point). Press ENTER
5. Press ENTER to reveal zero
You will need to repeat for each zero.
b2
- 4ac > 0 and not a perfect square
Academic
FACTORING FLOW CHART
Check for a GCF.
Polynomial with 4 or more terms
Factor by Grouping
Addition or Subtraction?
Binomial Factors
Binomial (2 terms)
Trinomial (3 terms)
Factor into Binomial Factors
(Use any method or shortcut you’ve learned)
Binomial Factors
Addition Subtraction
Done
Check for Difference of Squares
a2-b2=(a+b)(a-b)
Done
Sum of Squares
PRIME
Honors
FACTORING FLOW CHART
Check for a GCF.
Polynomial with 4 or more terms
Factor by Grouping
Addition or Subtraction?
Binomial Factors
Binomial (2 terms)
Trinomial (3 terms)
Factor into Binomial Factors
(Use any method or shortcut you’ve learned)
Binomial Factors
Addition Subtraction
Done
Check for Difference of Squares or Cubes
a2-b2=(a+b)(a-b) a3-b3=(a-b)(a2+ab+b2
)
Done
Check for Sum of Cubes
a3+b3=(a+b)(a2-ab+b2)
Quadratics Worksheet 1
Graph: plot the vertex and 4 more points (2 on each side of vertex)
Parent Function y = x2 1. y = (x – 2)
2 + 1 2. y = x
2 +6x+5
3. y = –(x + 1)2 + 3 4. y = 2x
2 –12x +13 5. y = (x)
2 – 2
Put the following in standard form f(x) = ax2 + bx + c. Name the vertex and axis of symmetry!
6. f(x) = (x – 3)2 + 4 7. f(x) = (x + 1)
2 – 3 8. f(x) = 2(x – 4)
2 – 3
Solve by Graphing:
9.
Solve by Graphing:
Parent Function y = x2 10. y = (x – 3)
2 – 1 11. y = -x
2 – 4x
x- intercepts: __________ _____________ _____________
12. y = –3(x + 4)2 + 3 13. y = 2x
2 – 4x 14. y = (x – 5)
2 – 2
x- intercepts: __________ _____________ _____________
Name the vertex of the graph _______________
Name the axis of symmetry ________________
What are the x-intercepts? _________________
Write the equation ________________________
Solve by factoring:
15. x – 2x – 15 = 0 16. z – 5z = 0
17. x + 6x = -9 18. 3q – 7q = 20
18. 9y = 49 19.
2c2
- 24c + 54 = 0
20.
25x2
- 4 = 0 21.
25x2
- 30x + 9 = 0
Solve by taking the square root:
22.
5a2
-15 = 0 23.
3 x - 2( )2
= 24
24.
1
5x - 4( )
2
= 6 24.
3x2
+ 42 = 0
Quadratic Worksheet 2
Identify the quadratic term, the linear term, and the constant term for each function.
1. f(x) = x2 + 14x + 49 2. f(x) = -3(2x + 1)
2
Graph each function. Name the vertex and the axis of symmetry.
3. f(x) = x2 – 10x + 25 4. f(x) = (x + 4)
2 – 6 5. f(x) = -(x – 1)
2 + 4
Vertex: __________ _____________ _____________
axis of sym: __________ _____________ _____________
Solve (i.e. find the x-intercepts) by graphing.
6. f(x) = -(x + 5)2 + 1 7. f(x) = x
2 + 2x 8. f(x) = 2(x + 3)
2 – 8
Vertex: __________ _____________ _____________
x- intercepts: __________ _____________ _____________
Solve each equation. Remember to set equal to zero. If there is a linear term you can solve
some by factoring. If there is no linear term solve by taking the square root.
9. x2 – 4x – 12 = 0 10. x
2 – 16x + 64 = 0
11. x2 + 25 = 10x 12. 9z = 10z
2
13. 7x2 – 4x = 0 14. x
2 = 2x + 99
15. 5w2 – 35w + 60 = 0 16. 3x
2 + 24x + 45 = 0
17. 15m2 + 19m + 6 = 0 18. 4x
2 + 6 = 11x
19. 36x2 = 25 20. 12x
3 – 8x
2 = 15x
21. 6x3 = 5x
2 + 6 x 22. 9 = 64x
2
Factor Worksheet Page 1 - Factor completely. 1. ax + ay + bx + by 2. 8x2 + 2xy + 12x + 3y 3. 6mn – 9m – 4n + 6 4. 2x2y + 6xy – x – 3 5. 4k + 12 + k2 + 3k 6. p2q + pq – 1 – p 7. 2ac + ad + 6bc + 3bd 8. 4r2s – 8rs – 3r + 6 9. z3 – 6 + 2z – 3z2 10. 3a – 5a2 – 6b + 10ab 11. 2uv – u2v – 6 + 3u 12. 6cd2 – 8cd – 9d + 12 13. 2e2f – 12ef + 3e – 18 14. 3ac + 3bc + ad + bd 15. 2cx + cy – 2dx – dy 16. bx4 – by4 + cx4 – cy4
17. r2 + 6rt + 9t2 – a2 – 2ab – b2 18. 4x2 + 4xy + y2 – 9a2 – 12at – 4t2 19. 6x3 + 9x – 4x2 – 6 20. 2xz – 6xy + 2yz – 6y2
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Quadratic Formula:
Solve using the Quadratic Formula
1.
3x2
+ 8x = 35 2.
3. 4.
5. 6.
Solve by taking the square root:
7. 3x2 = -81 7. 5x
2 + 18 = 3
8. (m – 2)2 = -16 10.
x =-b ± b
2- 4ac
2a
Complex Numbers We do NOT get a real number when we take the square root of a negative number. For example 9 is not a real number because there is no real number that can be squared to a get -9.
Imaginary numbers are used when there is a negative number under a square root. “i” is used to signify an imaginary number. The reason for the name "imaginary" numbers is that when these numbers were first proposed several hundred years ago, people could not "imagine" such a number.
i= 1 so … 4 = 41 =
i 4= 2i
i = 1 i5 = i9 = i13 = i2 = i6 = i10 = i14 = i3 = i7 = i11 = etc…. i4 = i8 = i12 = To simplify imaginary numbers with an exponent greater than 3:
1) Divide the exponent by 4 2) The remainder becomes the new exponent 3) Simplify
Examples: i13 i12 i94 i27
To simplify the square root of a negative number: 1) pull out the i 2) simplify the radical
Examples: 30 24 45
If two square roots with negative numbers are being multiplied: pull out the i BEFORE you multiply!
Examples: 10- 6- 28
Adding/Subtracting: combine like terms
Examples: (8 – 5i) + (2 + i) (4 + 7i) – (2 – 3i)
Multiplying with imaginary numbers: NEVER leave i2 in your answer!
Examples: (4 + 2i)(3 – 5i) (4 – i)(3 + 2i)
A complex number is any number that can be written in the standard form a + bi, where a and b are real numbers, and i= 1 .
real numbers are complex numbers with b=0 pure imaginary numbers are complex numbers with a=0
Complex numbers in equations: Find the values of x and y for which each equation is true. Examples: 4x – 3yi = 16 + 9i 6x + 2yi = -18 + 3i Every complex number has a complex conjugate. The complex conjugate of a + bi is a - bi . The conjugate of 3 + 5i is 3 – 5i. What happens when you multiply conjugates? Examples: (2 + i)(2 – i) (3 + 5i)(3 – 5i) Conjugates can be used to rationalize the denominator of a fraction:
Simplify:
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x2
- 6x + 9 = 25
(x - 3)2
= 25
(x - 3)2
= ± 25
(x - 3) = ±5
x = 3 ± 5
x = 8,-2
+
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+ + + + 43++
x2
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+
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53++
x2
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+ + + + 63++
x2
- 8x _____ = ( )2
+
+
73++
x2
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+ + + + 83++
x2
-12x _____ = ( )2
+
+
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+ + + + :3++
x2
+ 8x _____ = ( )2
+
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13. 14. 15.
16. 17.
Worksheet 8.3 Write each equation in the form f(x) = (x – h)2 + k. Then name the vertex and the axis of symmetry for the graph of each function. 1) f(x) = x2 – 10x + 25 2) f(x) = x2 + 12x + 36 3) f(x) = x2 + 2 4) f(x) = x2 – 6x 5) f(x) = x2 – 3x – 1 6) f(x) = x2 – 2x – 1 Draw the graph of each equation: 7) f(x) = (x – 3)2 + 2 8) f(x) = (x + 5)2 – 1
y
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9) f(x) = -x2 + 2x + 6 10) f(x) = x2 – 4x + 7
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Worksheet 8.4 Write each equation in the form f(x) = (x – h)2 + k. Then name the vertex, the axis of symmetry, and the direction of opening for the graph of each function. 1) f(x) = -6x2 2) f(x) = -2x2 – 16x – 32
3) f(x) = 6432 2 ++ xx 4) f(x) = 2x2 + 16x + 29
5) f(x) = -9x2 + 12x – 4 6) f(x) = -3x2 + 6x – 5
7) f(x) = 17643 2 ++ xx 8) f(x) = 7x2 – 56x + 116
Draw the graph of each equation: 9) f(x) = -2x2 + 1 10) f(x) = -3x2 + 6x – 5
y
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Solving Quadratic Equations
Solve using any method.
1. 2x2 = 16
2. 4x2 + 8 = 0
3. 3x2 + 8x + 4 = 0
4. 9x2 + 15 = 0
5. 3x2 + 8 = 10
6. 2y2 + 2y – 24 = 0
7. b2 - 12b = 2b – 45
8. x2 = 8x + 20
9. 25x2 = -4
10. 5x2 + 6x – 12 = -4
11. 4x2 = 9
12. 2x2 + 12 = 0
13. 3x2 - 7x = 6
14. 2x2 = 12x – 16
15. x2 + 6x = 40
16. 15x2 = -10
17. x2 – 3x + 20 = 38
18. 15x2 + 8 = 5
19. 3n2 – 6n – 45 = 0
20. 5x2 – 12 = 18
21. 9x2 – 3x = 0
22. 3x2 – 8x = 0
23. 8x2 – 12 = -15
24. y2 - 7y = 30
25. x2 -7x + 10 = 0
26. 4x2 = 6x
27. 3x2 +4x – 12 = 3
28. 6x2 +17x + 5 = 0
29. 4y2 = -11y – 6
30. 6x2 = 3 - 7x
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Sum & Product of Roots Worksheet (Honors)
Solve each equation, then find the sum and product of the roots to check your solutions.
1. x2 – 7x + 4 = 0 2. x2 + 3x + 6 = 0
3. 2n2 + 5n + 6 = 0 4. 7x2 – 5x = 0
5. 4r2 – 9 = 0 6. –5x2 – x + 4 = 0
7. 3x2 + 8x = 3 8.
Write the quadratic equation, in standard form, that has the given roots
9. 7, -3 10.
11. 12.
13. 14. 7 – 2i , 7 + 2i
15. 8i, -8i 16.
17. 18.
Find k such that the number given is a root of the equation.
19. 7; 2x2 + kx – 21 = 0 20. –2; x2 – 13x + k = 0
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Quadratic Modelling Word Problems
Example:
1. A ball is thrown upward into the air with an initial velocity of 80 ft/sec. The formula h(t) = 80t – 16t2 give its height h(t) after t seconds.
a) What is the height of the ball after 2 seconds? b) What is the maximum height of the ball? c) How long does it take the ball to reach its maximum height? d) How long is the ball in the air?
Practice:
1.Awaterballooniscatapultedintotheairsothatitsheight,inmeters,aftertsecondsis
modeledbytheequationh(t)=‐4.9t2+27t+2.4
a) Howhighistheballoonafter1second?
b) Whatisthemaximumheightoftheballoon?
c) Howlongdoesittaketheballoontoreachitsmaximumheight?
d) Whenwilltheballoonhitthegroundandburst?
2.Atafestival,pumpkinsarelaunchedwithlargecatapultsandaircannons.On
onelaunch,theheightofapumpkininfeetabovethegroundaftertsecondsis
modeledbyh(t)=‐16t2+100t+12
a) Findthemaximumheightofthepumpkin.
b) Whendidthepumpkinreachitsmaxheight?
c) Whendidthepumpkinhittheground?
d) Whatwastheheightofthepumpkinafter5seconds?
e) Whenwasthepumpkinataheightof100feet?
ANSWER: 1) a. 24.5m b. 39.6m c. 2.8 sec d. 5.6 sec
2) a. 168.25 sec b. 3.125 sec c. 6.4 sec d. 112 ft e. 1.06 & 5.19 secs
3.Anobjectisthrownupwardintotheairwithaninitialvelocityof128feetper
second.TheformulaH(t)=128t–16t2givesitsheightafter“t”seconds.
a)Whatistheheightafter2seconds?
b)Whatisthemaximumheightreached?
c)Forhowmanysecondswilltheobjectbeintheair?
4.Supposeyouaretossinganappleuptoafriendonathird‐storybalcony.After
tseconds,theheightoftheappleinfeetisgivenbyh(t)=‐16t2+38.4t+.96.
Yourfriendcatchestheapplejustasitreachesitshighestpoint.Howlongdoes
theappletaketoreachyourfriend,andatwhatheightabovethegrounddoes
yourfriendcatchit?
5.Thebarber’sprofit(p)eachweekdependsonhischarge(c)perhaircut.Itis
modeledbyp=‐200c2+2400c–4700.Whatpriceshouldhechargeforthe
largestprofit?Whatisthemaximumprofit?
6. Thepathofabaseballafterithasbeenhitismodeledbythefunction
h(d)=‐.0032d2+d+3,wherehistheheightinfeetofthebaseballanddisthe
distanceinfeetthebaseballisfromhomeplate.Whatisthemaximumheight
reachedbythebaseball?Howfaristhebaseballfromhomeplatewhenit
reachesitsmaximumheight?
ANSWER: 3) a. 192 ft b. 256 ft c. 8 sec 4) 1.2 sec , 24 ft 5) $6& $2500 6) 81 ft& 156 ft
Quadratic Equations Review #1
Solve each equation by factoring.
1. x2 ! 4x ! 32 = 0 2. 4x
2 + 20x = 0 3. d
2 ! 29d = -100 4. 18x
2 + 29x + 3 = 0
Solve each equation by completing the square.
5. x2 + 4 = 8x 6. x
2 !
5x = 8 7. 2x
2 ! 12x = 8 8. 4x
2 !12x = 16
Solve each equation by using the quadratic formula.
9. x2 + 2x = 7 10. 2x
2 ! 12x + 5 = 0 11. 2x ! 5x
2 + 3 = 0 12. 6x
2 ! 3x + 2 = 0
Solve each equation by using any method.
13. 3x2 + 6x +3 = 0 14. x
2 + 6x = 4 15. 2x
2 + x !1 = 3 16. 3x
2 + 2 = -7x
17. r2 = 3r + 70
18. (x ! 3)
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2 – 8x + 9 = 4 20. 4x
2 + 8x = -3
Write the equation of the parabola with the given info:
21. Focus (-1, 6) Directrix y=0 22. Focus (3,-2) Directrix y=-4
Given a = -3 + 2i and b= 4-5i
23. Find a+ b 24. Find a – b 25. Find the product of a and b
26. Find 2a – 3b 27. Find a2 – b
2
Use your calculator to answer the following questions:
28. A ball is thrown upward vertically with an initial speed of 96 feet per second. The
equation h = 96t – 16t2 gives the height of the ball in t in seconds. What is the
maximum height reached by the ball? When will the ball be 128 feet above its
starting point?
29. Terry has 200 yards of fencing to enclose a rectangular garden on three sides. The
fourth side will be the side of the house. What dimensions of the garden will
maximize the area?
(HONORS) Write the quadratic equation with the given solutions
30. 3, -8 31. -5, 32.
33. 34. 35.
(HONORS) Solve by factoring.
36. x4 ! 6x
2 + 5 = 0 37. a
3 ! 81a = 0 38. 39.
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More Factoring Worksheet #1 FACTOR COMPLETELY! 1. yxyxyx 42538 5123 −+− 2. 3176 2 −+ aa 3. xxx 284914 23 −+ 4. 127 3 +a 5. 2452 +−− xx 6. 181512 2 −− xx 7. 22 20baba −− 8. 8481 yx − 9. 164 4 −a 10. 92416 2 ++ aa 11. 322 456 yxyyx ++− 12. )8()8( ybyx −+− 13. ( ) ( )yxyxx 334 2 −−− 14. 42025 2 ++ yy 15. 3214 48 −− nn xx 16. 36 64125 yx − 17. 362743 22 −+− kyky 18. 22 9374 yxyx +− 19. xx 216 4 +− 20. 223 18248 abbaa −+−
More Factoring worksheet #2 FACTOR COMPLETELY!
1. aab 102525 24 +−− 2. 33 271000 pn +
3. 22224 4736 bbaba ++− 4. 3214 48 −− nn xx
5. ( ) ( ) 1323 2 ++++ yxyx 6. 144 2222 −−+ baba
7. ( ) ( )22 +−+ xax 8. baba 3322 +−−
9. xx 25016 4 +− 10. 21632 24 +− cc
11. 96 24 ++ aa xx 12. 481 y−
13. 63 zy a + 14.
2222 916364 yxyx −+−
15. 22224 4736 bbaba ++− 16. zzx 81622 +−−
17. 22325 72262 xyyxyx +− 18. 645 2 −+ xx
19. ( ) 44 3 −+ nm 20. 66 yx −
More Factoring Worksheet #3 FACTOR COMPLETELY! 1. 2y2 – 242 2. 2x3y – x2y + 5xy2 + xy3 3. 8m3 – 1 4. b4 – 81 5. 4a2 + a – 3 6. 5x2 – 40x + 80 7. 21 – 7x + 3y - xy 8. x2 + 2x - xy - 2y 9. 4x6 – 4x2 10. 36a3b2 – 210ab4 + 66a2b3 11. 45x2 – 80y2 12. 4a2 + 12ab + 9b2 – 25c2 13. x2 – 10xy + 25y2 – 16 14. x2y2 – 3x2 -4y2 + 12 15. 64a4 + 27a 16. 9a2 + 16b2 17. 6p2m – 6mpq + 21mp -21mq
More Factoring Worksheet #4 FACTOR COMPLETELY! 1. ( ) ( )439432 −+− kky 2. 3214 48 −− nn xx 3. xx 216 4 +− 4. 645 42 −+ xx 5. 32 22048 xxx +− 6. 36 27125 yx + 7. 223 18248 abbaa −+− 8. 44 16yx − 9. 62 2 −− xx 10. 42025 2 ++ yy 11. 22 9374 yxyx +− 12. ( ) ( )yxyxx 334 2 −−− 13. ( ) ( )222 52 yxba −−+ 14. 82 3 −x
Adapted'from'Core-Plus'Mathematics,'Course'3,'Unit'5,'Lesson'1'!Introduction*to*Polynomial*Expressions*and*Functions*!Suppose!you!work!for!a!company!that!designs,!builds,!and!tests!rides!for!amusements!parks.!!Your!team!is!in!charge!of!designing!a!long!roller!coaster.!!One!morning,!your!team!is!handed!sketches!that!show!ideas!for!two!sections!of!the!new!roller!coaster.!!!!
!!Your!task!is!to!find!algebraic!functions!with!graphs!that!match!the!two!sketches.!!The!functions!will!be!useful!in!checking!safety!features!of!the!design,!like!estimated!speed!and!height!a!various!points!of!the!track.!!They!will!also!be!essential!in!planning!manufacture!of!the!coaster!track!and!support!frame.!!!!Think*About*This*Situation:*a.!!What!familiar!functions!have!graphs!that!match!all!or!parts!of!the!design!sketches?!
!!!!b.!!!What!strategies!could!you!use!to!find!functions!with!graphs!that!model!the!sketches?!
!!!!c.!!What!do!you!think!are!the!key!points!on!each!sketch!that!should!be!used!in!finding!a!function!model!for!the!graph!pattern?!!
!!!
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POLYNOMIALS*
You!have!already!studied!linear!and!quadratic!functions.!!These!functions!are!from!a!larger!class!called!polynomial)functions.!!Cubic!and!quartic!functions!are!also!polynomial!functions.!!!A!monomial*is!an!expression!that!is!either!a!real!number,!a!variable,!or!the!product!of!real!numbers!and!variables.!!!A!polynomial!is!a!monomial!or!the!sum!of!monomials!!
A!polynomial)function!is!any!function!with!a!rule!that!can!be!written!in!the!form:!P(x)'=''anxn'+'an-1xn-1'+'…''+'a1x'+'a0''''''where!!an!≠!0,!!coefficients!(an,!…,!a0)!are!real!numbers,!and!exponents!are!nonnegative!integers.!
Any!algebraic!expression!in!the!form!anxn'+'an-1xn-1'+'…''+'a1x'+'a0''is!called!a!polynomial)expression.!!!!
One!of!the!most!important!characteristics!of!any!polynomial!function!or!expression!is!its!degree.!!The!degree)of)a)polynomial!is!the!greatest!exponent!of!the!variable.!!A!constant!polynomial!has!degree!0,!a!linear!polynomial!has!degree!1,!a!quadratic!polynomial!has!degree!2,!a!cubic!polynomial!has!degree!3,!a!quartic!polynomial!has!degree!4,!and!a!quintic!polynomial!has!degree!5.!!A!nonzero!constant!is!a!polynomial!of!degree!0!(ex.!y=5).!!The*leading)coefficient!of!a!polynomial!is!the!coefficient!of!the!term!with!the!highest!degree!In*standard)form*the!terms!of!a!polynomial!are!written!in!descending!order!by!degree.!!No!two!terms!have!the!same!degree!(since!like!terms!have!been!combined).!Polynomials!are!classified)by:!
! the!number!of!terms!(1=monomial,!2=binomial,!3=trinomial,!4+!=polynomial)!! the!degree!!
!
These!are!polynomials!in!one!variable!in!standard!form:!! ! ! ! !1.!!!6x4!+!3x2!+!4x!–!8!!!!(polynomial,!quartic)!!!!!!!!!!!2.!!x!+!8!!(binomial,!linear)! !3.!!!–5! !!!(monomial,!constant)! ! !!!!!!!!!!!!!!4.!!x5!–!3x3!–!7!!(trinomial,!quintic)! !! ! !!!!These!are*Not!polynomials!in!one!variable:!1.!!9x3y5!+!2x2y6!–!4!!!!(2!variables)! ! !!!!!!!!!2.!!x!d3!!+!4x2!–!1!(negative!exponent)! !3.!!
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5x 7 + 3x 4 +2x!!(variable!in!denominator!means!a!negative!exponent)!
!Classify*the*following*polynomials*by*degree*&*number*of*terms*(use*words):*1.!!!f(x)!=!7x5!+!3x!–!10!!!!!!!!!!!!!!! ! !
2.!!!g(x)!=!2x7!+!3x4!+!x2!d!5x!–!10!!!!!!
3.!!!h(x)!=!d5x3!d!4x!!
4.!!!k(x)!=!2x!
5.!!!m(x)!=!!3x5!+!8x6!d!11x2!d!5x3!+!7x!d!1!!!!!!!!!!
GRAPHS of POLYNOMIAL FUNCTIONS:
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END BEHAVIOR: Is the behavior of the graph as x approaches +! or -!
If the degree is EVEN, both ends have the SAME behavior
! If the leading coefficient is positive, both ends are up
! If the leading coefficient is negative, both ends are down
If the degree is odd, the ends have OPPOSITE behavior
! If the leading coefficient is positive, the right end is up, left down
! If the leading coefficient is negative, the right end is down, left up
Leading
Coefficient Degree Example x " - ! x " !
+ even f(x) = x2 f(x) " ! f(x) " !
- even f(x) = -x2 f(x) " -! f(x) " -!
+ odd f(x) = x3
f(x) " -! f(x) " !
- odd f(x) = -x3
f(x) " ! f(x) " -!
The Fundamental Theorem of Algebra says that every polynomial with degree greater
then zero has at least one complex root. An extension of this theorem says that:
A polynomial of degree n has exactly n complex roots
In other words …. the degree of a polynomial = # of zeros/roots/solutions
Ex. x3 + 4x2 + 4x = 0 has 3 zeros Ex. x4 – 10x2 + 9 = 0 has 4 solutions
! solutions, zeros, and roots are the values of x which give y = 0
! complex roots means real and/or imaginary
o complex numbers have the form a + bi
! ‘n’ counts multiple roots the number of times they occur
o multiplicity is the number of times a zero occurs
! imaginary roots always come in conjugate pairs (a + bi, a – bi)
! each x-intercept represents a real root of the polynomial equation
! a polynomial function with odd degree must have at least 1 real root
o the graph must cross the x-axis at least once (think about the end behavior)
! a polynomial function with even degree will have either no real roots or an even
number of real roots
o the graph may or may not cross the x-axis, but if it does it will cross an even
number of times (think about the end behavior)
! every polynomial of degree n > 0 can be written as the product of a constant k
and n linear factors. P(x) = k(x – r1)(x – r2) (x – r3) ….(x – rn)
! to find zeros write the polynomial in factored form and set each factor = 0
! for polynomial P(x), if a is a zero then P(a) = 0
When finding the zeros of polynomials REMEMBER:
! #zeros = degree of polynomial = # of factors
! if a is a zero then (x-a) is a factor
! when you divide a polynomial by one of it’s factors the remainder is 0
! you can use division to break a polynomial down into its factors (just like you do
with numbers)
! for quadratics you have multiple tools for finding the zeros (factor, complete
the square, quadratic formula, graphing)
Finding the zeros of polynomials
Find the zeros of the polynomials and state the multiplicity of each zero:
Examples: 1. f(x) = 5( x - 3)2 (2x + 5) (x + 2)3
Set each factor = 0 x – 3 = 0 so x = 3, multiplicity 2 (there are 2 (x-3) factors) 2x + 5 = 0 so x = -5/2 x+2=0 so x = -2 multiplicity 3 (there are 3 (x+2) factors)
The zeros are 3 multiplicity 2, -5/2, and -2 multiplicity 3 2. f(x) = (5x2 - 19x - 4) (x2 – 49) The polynomial is not completely factored, so first factor completely. f(x) = (5x+1)(x – 4)(x + 7)(x – 7) Now you can set each factor = 0
5x + 1 = 0 so x = -1/5 x-4 = 0 so x = 4 x + 7 = 0 so x = -7 x – 7 = 0 so x = 7 The zeros are -1/5, 4, -7, and 7.
Practice: 3. f(x) = ( x + 5)3 ( 2x – 3 )
4. f(x) = 2x5 – 12x4 – 14x3
5. f(x) = ( 16x2 – 49)4 (x2 + 25)
6. f(x) = (x2 + 3x – 10)2 (x2 – 9)
7. f(x) = 2x2 (x - 1)3 (x + 2)
8. f(x) = x (x-1)2 (5x + 2)
Graphing Polynomial Functions Worksheet
To graph a polynomial function: 1. Find the zeros of the function. Remember real zeros = x-intercepts so
graph these points on the x-axis. 2. Find the y-intercept (value of y when x=0) 3. Determine the end behavior of the function based on the degree and
the leading coefficient. 4. Using the zeros, end behavior, and y-intercept to make a smooth curve.
Example: y = ½(x -2)2(x+1)(x-4) 1. zeros: 2 multiplicity 2 (so this is a bounce), -1, 4 2. y-intercept (plug in 0 for x): y = ½(0-2)2(0+1)(0-4) = -8 so (0, -8) 3. end behavior: degree is 4, +coefficient so ! ! 4. plot the zeros, y-intercept, end behavior & make a smooth curve
Graph each function. USE GRAPH PAPER! 1.
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y = −2 x 2 − 9( ) x + 4( ) 2. y = (x2 - 4)(x+3)
3. y = -1(x2-9)(x2-4) 4.
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y =14x + 2( ) x −1( )2
5.
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y =15x − 3( )2 x +1( )2 6.
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y = x +1( )3 x − 4( )
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Synthetic Division when the coefficient of x in the divisor ≠ 1 HONORS
Divide:
Step 1: Factor out the coefficient of x in the denominator.
4𝑥 − 8𝑥 − 𝑥 + 52(𝑥 − 1
2)
12
4𝑥 − 8𝑥 − 𝑥 + 5𝑥 − 1
2
Step 2:, Set up the synthetic division, ignoring the ½ that was factored out.
Step 3: Once the problem is set up correctly, bring the leading coefficient (first number) straight down.
Step 4: Multiply the number in the division box with the number you brought down and put the result in the next column.
Step 5: Add the two numbers together and write the result in the bottom of the row.
Step 6: Repeat steps 3 and 4 until you reach the end of the problem.
Step 7: Multiply everything by the ½ which was factored out in Step 1.
4𝑥 − 6𝑥 − 4 +
2𝑥 − 3𝑥 − 2 + 32𝑥 − 1
is the final answer
Dividing Polynomials - EXAMPLES !Dividing!by!a!monomial!1.!(N30x3y!+!12x2y2!–!18x2y)!÷!(N6x2y)! ! !!Divide)using)Long)Division)2.!(6x2!–!x!–!7)!÷!(3x!+!1)! ! !!!!!!!!!!!!!!!!3.!! !(4x2!–!2x!+!6)(2x!–!3)N1! !! !!!!!!4.!(4x3!–!8x2!+!3x!–!8)!÷!(2x!–!1)!!!!!!!!!!!!!!!!!5.!!(2x3!–!3x2!–!18x!–!8)!÷!(x!–!4)!!!!!!6.!!
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(2x 4 + 3x 3 + 5x −1) ÷ (x 2 − 2x + 2)!!!!!!!! ! !!Divide)using)Synthetic)Division)7.!(2x2!+!3x!–!4)!÷!(x!–!2)!!!!!!!!!!!!!!!!!!!!!!!!!8.!!(x4!–!3x3!+!5x!–!6)!÷!(x!+!2)!!!!!9.!(2x3!+!4x!–!6)!÷!(x!+!3)!!!!!!!!!!!!!!!!!!!!!!!!!10.!(x4!–!2x3!+!6x2!–8x!+!10)!÷!(x!+!2)!!
)
(HONORS)!11.)(6x4!–!x3!+!3x!+!5)!/!(2x!+!1)!
Polynomial Division Worksheet
Divide using Synthetic Division
1. (3y3 + 2y
2 – 32y + 2) / (y – 3)
2. (2b3 + b
2 – 2b + 3) / (b + 1)
3. (2c3 – 3c
2 + 3c – 4) / (c – 2)
4. (3x3 – 2x
2 + 2x – 1) / (x – 1)
5. (t4 – 2t
3 + t
2 – 3t + 2) / (t – 2)
6. (3r4 – 6r
3 – 2r
2 + r – 6) / (r + 1)
7. (z4 – 3z
3 – z
2 – 11z – 4) / (z – 4)
8. (2b3 – 11b
2 + 12b + 9) / (b – 3)
9. (6s3 – 19s
2 + s + 6) / (s – 3)
10. (x3 + 2x
2 – 5x – 6) / (x – 2)
11. (x3 + 3x
2 – 7x + 1) / (x – 1)
12. (n4 – 8n
3 + 54n + 105) / (n – 5)
13. (2x4 – 5x
3 + 2x – 3) / (x – 1)
14. (z5 – 6z
3 + 4x
2 – 3) / (z – 2)
15. (y4 + 3y
3 + y – 1) / (y + 3)
Divide using long division:
16. (4s4 – 5s
2 + 2s + 3) / (2s – 1)
17. (2x3 – 3x
2 – 8x + 4) / (2x + 1)
18. (4x4 – 5x
2 – 8x – 10) / (2x – 3)
19. (6j3 – 28j
2 + 19j + 3) / (3j – 2)
20. (y5 – 3y
2 – 20) / (y – 2)
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Remainder Theorem: The value of the polynomial p(x) at x=a is the same
as the remainder you get when you divide that polynomial p(x) by x – a.
! To evaluate a polynomial p(x) at x = a, use synthetic division to divide
the polynomial by x = a. The remainder is p(a).
Use the Remainder Theorem and synthetic division to find f(4) where
f(x) =
The Remainder Theorem tells us that if we use synthetic division and divide
f(x) by (x-4), the remainder will be equal to f(4).
The remainder is 127. So f(4) = 127.
Factor Theorem: p(a) = 0 if and only if x – a is a factor of p(x).
! If you divide a polynomial by x = a and get a zero remainder, then, not
only is x = a a zero of the polynomial, but x – a is also a factor of the
polynomial.
Determine whether x + 4 is a factor of each polynomial.
Note: synthetic division can be used instead of long division !
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Practice 6-3 Dividing Polynomials
Determine whether each binomial is a factor of x3 ± 3x2 – 10x – 24.
1. x + 4 2. x - 3 3. x + 6 4. x + 2
Divide using synthetic division.
5. (x3 - 8x2 + 17x - 10) ! (x - 5) 6. (x3 + 5x2 - x - 9) ! (x + 2)
7. (-2x3 + 15x2 - 22x - 15) ! (x - 3) 8. (x3 + 7x2 + 15x + 9) ! (x + 1)
9. (x3 + 2x2 + 5x + 12) ! (x + 3) 10. (x3 - 5x2 - 7x + 25) ! (x - 5)
11. (x4 - x3 + x2 - x + 1) ! (x - 1) 12.
13. (x4 - 5x3 + 5x2 + 7x - 12) ! (x - 4) 14. (2x4 + 23x3 + 60x2 - 125x - 500) ! (x + 4)
Use synthetic division and the Remainder Theorem to find P(a).
15. P(x) = 3x3 - 4x2 - 5x + 1; a = 2 16. P(x) = x3 + 7x2 + 12x - 3; a = -5
17. P(x) = x3 + 6x2 + 10x + 3; a = -3 18. P(x) = 2x4 - 9x3 + 7x2 - 5x + 11; a = 4
Divide using long division. Check your answers.
19. (x2 - 13x - 48) ! (x + 3) 20. (2x2 + x - 7) ! (x - 5)
21. (x3 + 5x2 - 3x - 1) ! (x - 1) 22. (3x3 - x2 - 7x + 6) ! (x + 2)
Use synthetic division and the given factor to completely factor eachpolynomial function.
23. y = x3 + 3x2 - 13x - 15; (x + 5) 24. y = x3 - 3x2 - 10x + 24; (x - 2)
Divide.
25. (6x3 + 2x2 - 11x + 12) ! (3x + 4) 26. (x4 + 2x3 + x - 3) ! (x - 1)
27. (2x4 + 3x3 - 4x2 + x + 1) ! (2x - 1) 28. (x5 - 1) ! (x - 1)
29. (x4 - 3x2 - 10) ! (x - 2) 30.
31. A box is to be mailed. The volume in cubic inches of the box can be expressed as the product of its three dimensions:V(x) = x3 - 16x2 + 79x - 120. The length is x - 8. Find linearexpressions for the other dimensions. Assume that the width is greater than the height.
(3x3 2 2x2 1 2x 1 1) 4 ax 1 13b
ax4 1 53x
3 2 23x
2 1 6x 2 2b 4 ax 2 13b
Name Class Date
Lesson 6-3 Practice Algebra 2 Chapter 64
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16. Factor f(x) = 9x3 + 6x2 – 3x if you know (x+1) is a factor.
17. Factor f(x) = x3 –2x2 – 9x + 18 if you know (x+3) is a factor.
18. Factor y = x3 – 4x2 – 3x +18 if you know that (x+2) is a factor.
19. Show that –3 is a zero of multiplicity 2 of the polynomial function
P(x)= x4 + 7x3 + 13x2 – 3x –18 and express P (x) as a product of linear factors.
20. Show that –1 is a zero of multiplicity 4 of the polynomial function
f(x)= x5 + x4 – 6x3 – 14x2 – 11x –3 and express f (x) as a product of linear factors.
21. Find a polynomial function of degree 4 such that both –2 and 3 are zeros of multiplicity 2.
22. Find a polynomial function of degree 5 such that –2 is a zero of multiplicity 3 and 4 is a zero of multiplicity 2.
23. Determine k so that that f(x ) = x3 + kx2– kx +10 is divisible by x +3.
24. Find k so that when x3 – x2 –kx + 10 is divided x –3 , the remainder is –2.
25. Find k so that when x3 – kx2– kx +1 is divided by x-2, the remainder =0
26. Determine k so that that f(x ) = 2kx3 + 2kx - 10 is divisible by x - 2.
27. SOLVE x3 + 4x2 – 5x = 0 completely.
#28-35 HONORS 28. SOLVE x4 + 7x2 –18 = 0 completely.
29. Determine all values of k so that f(x ) = k2x2 – 4kx +3 is divisible by x – 1.
39. Find the remainder if the polynomial 3x100 + 5x85 – 4x38 +2x17 – 6 is divided by x+1
31. Write a cubic equation having zeros 2, 43
and -1.
32. Write the quartic equation having zeros 2i and 3 – i .
33. Write the cubic equation having zeros 32
and 2 + 3i
34. SOLVE 2x4 - 17x3 + 47x2 – 32x – 30 = 0 given that 3+i is a root.
35. SOLVE x4 - x3 + x2 + 9x – 10 = 0 knowing 1 – 2i is a root.
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HONORS
EXAMPLES: Find the possible rational roots, then find all the zeros.
1.
3x3
- x2
-15x + 5 = 0 2.
x4
- 5x3
+ 9x2
- 7x + 2 = 0
PRACTICE:
1. Solve
2. Solve
3. Find all the zeros of
f x( ) = x4
- x3
+ 2x2
- 4x - 8
4. Find all the roots of
5. Find all the zeros of
6. Find all the solutions of
0 = 15x4
+ 68x3
- 7x2
+ 24x - 4
Rational Root Theorem Worksheet
Find all the roots.
1.
p x( ) = x4
+ 5x3
+ 5x2
- 5x - 6 2.
p x( ) = x3
- 5x2
- 4x + 20
3.
p x( ) = x4
- 5x3
+ 9x2
- 7x + 2 4.
p x( ) = x3
- 2x2
- 8x
5.
p x( ) = x3
+ 7x2
+ 7x -15 6.
p x( ) = 2x3
- 5x2
- 28x +15
7.
p x( ) = x3
- 7x - 6 8.
p x( ) = x4
+ 2x3
- 9x2
- 2x + 8
Polynomial Modelling Word Problems
When an equation is NOT given: 1. Define your variable(s) 2. If needed draw a picture. 3. Write an equation(s) to solve the problem. 4. State the solution. 5. Explain in words how you found the solution.
EXAMPLES: 1. The length of a rectangular pool is 4 yd longer than its width. The area of the pool is 60 yd. What are the dimensions of the pool? (6 x 10 yds) 2. A rectangle has a perimeter of 52 inches. Find the dimensions of the rectangle with maximum area. (13 x 13 in) 3. Find two consecutive negative integers whose product is 240. (-15 & -16) 4. Find two numbers who sum is 20 and whose product is a maximum (10 & 10).
Polynomial Word Problem Worksheet 1. Find two consecutive positive integers whose product is 462. 2. Find two numbers who difference is 8 and whose product is a minimum. 3. A rectangle has a perimeter of 48 inches. Find the dimensions of the rectangle with
maximum area.
4. Find the negative integer whose square is 10 more than 3 times the integer. 5. One side of a rectangular garden is 2 yd less than the other side. If the area of the
garden is 63 yd2, find the dimensions of the garden.
ANSWERS: 1) 21 & 22 2) 4 & -4 3) 12 x 12 in 4) -2 5) 7 x 9 yds.
6. Find two numbers who sum is -12 and whose product is a maximum. 7. Find 2 numbers whose sum is 36 and whose product is a maximum. 8. Find 2 numbers whose difference is 40 and whose product is a minimum. 9. A rectangle has a perimeter of 40 meters. Find the dimensions of the rectangle with the maximum area. 10. Nick has 120 feet of fencing for a kennel. If his house is to be used as one
side of the kennel, find the dimensions to maximize the area. ANSWERS: 6) -6 & -6 7) 18 & 18 8) 20 & -20 9) 10 x 10 10) 60 x 30
WordProblems HONORS
1. AtrolleycarriestouriststhroughthehistoricdistrictofBostonserving300
customersaday.Thechargeis$8perperson.Thecompanywouldlose20
passengersadayforeach$1increaseinfare.Whatchargewouldbethemost
profitable,andwhatisthemaximumprofit.
2.Aclothingstoresells40pairsofjeansdailyat$30each.Theownerfiguresthat
foreach$3increaseinprice,2fewerpairswillbesoldeachday.Whatprice
shouldbechargedtomaximizeprofit?
3.Asquare,whichis2in.by2in.,iscutfromeachcornerofarectangularpieceof
metal.Thesidesarefoldeduptomakeabox.Ifthebottommusthavea
perimeterof32in.,whatwouldbethelengthandwidthformaximumvolume?
Whatisthemaximumvolume?
4. Judgingbyhispastperformanceonmathexams,Rosscanestimatethegradehe
willreceiveonamathexamusingthefunctiony=‐t2+8t+78,wheret
representsthenumberofhoursspentstudying.Whatistheleastnumberof
hourshehastostudyinordertoreceiveagradeof90?
5.Findanumberwhosesquareis55greaterthan6timesthenumber.
6. Irmak’sItemsmakescustomhand‐paintedT‐shirts.Thecompany’sprojected
annualrevenuecanbemodeledbythefunctionR(x)=12x2+120x+111,where
xisthenumberofitemsproducedinhundreds.Thecosttoproducetheseitems
canbemodeledbythefunctionC(x)=4x2+35x+91.Ifthecompanysellsevery
itemthatitproduces,itwillmakeaprofitof$222.Howmanyitemsdidthe
companysell?
7. Arectangleis6cmlong5cmwide.Wheneachdimensionisincreasedbythe
samenumbertheareaistripled.Findthenumbereachdimensionwas
increasedby.
HONORS
8. Arectangularfieldwitharea5000m2isenclosedby300moffencing.Findthe
dimensionofthefield.
9. Arectangularanimalpenwitharea1200m2hasonesidealongabarn.Theother
threesidesareenclosedby100moffencing.Findthedimensionsofthepen.
10.Apositivenumberisonemorethanitsreciprocal.Findthenumber.
11.Twopositivenumbershaveasumof5andaproductof5.Findthenumbers.
12.A5inby7inphotographissurroundedbyaframeofuniformwidth.Thearea
oftheframeequalstheareaofthephotograph.Findthewidthoftheframe.
13. Awalkwayofuniformwidthhasarea72m2andsurroundsaswimmingpool
thatis8mwideand10mlong.Findthewidthofthewalkway.
14.Thetotalsurfaceareaoftherectangularsolidshownis36m2.Findthevalueofx.
15.Aboxwithheight(x+5)cmhasasquarebasewithsidexcm.Asecondbox
withheight(x+2)cmhasasquarebasewithside(x+1)cm.Ifthe2boxes
havethesamevolume,findx.
16.Aboxwithasquarebaseandnolidistobemadefromasquarepieceofmetal
bycuttingsquaresfromthecornersandfoldingupthesides.Thecut‐offsquares
are5cmonaside.Ifthevolumeoftheboxis100cm3,findthedimensionsof
theoriginalpieceofmetal.
x
2x
x +2
(x y)2 x 2 2xy y 2
(x y)3 x 3 3x 2y 3xy 2 y 3
(x y)4 x 4 4x 3y 6x 2y 2 4xy 3 y 4
Pascal’s Triangle to Expand Binomials This figure shows a portion of Pascal’s triangle completed through row 6. The pattern used to build Pascal’s triangle is infinite. Note how the rows and diagonals are labeled. You can locate an element in the triangle by giving its row number and diagonal number. For example, the element “20” is at row 6 and diagonal 3.
Pascal’s triangle can be used to expand powers of binomials. Consider: Notice the relationship between the coefficients for each expanded term and a row in Pascal’s triangle.
To expand
(x y)n using Pascal’s triangle:
The first term is
xn and the last term is
yn In successive terms, the exponent of
x decreases by 1 and the exponent of
y increases by 1. Use row n for the coefficients of the terms (note that the coefficient of the 2nd
term is always equal to n, the element at row n and diagonal 1) Note: the expansion has n+1 terms for each term: the degree of x + the degree of y = n the coefficients are symmetric
Pascal’s Triangle to Expand Binomials - Worksheet
Example 1:
(x y)6
Example 2:
(2x y)7
Example 3:
(x 2y)4
Homework: Expand using Pascal’s Triangle:
1. ( y – 3)5 2. ( x-y)10 3. ( 2x + y)6 4. ( 4x – y)3
5. ( 3a – 5b)4
6. ( a – b2)5
7. ( x2- 3y )4
8. (a3 – 2b2)6
9. (x + 1/x)6
WRITING EQUATIONS OF POLYNOMIALS Write the equation from the graph: 1. 2. 3.
1. _________________________ 2. __________________________ 3. ___________________________
4. 5. 6.
4. _________________________ 5. __________________________ 6. ___________________________
7. 8. 9.
7. _________________________ 8. __________________________ 9. ___________________________
Polynomial)Review)#1)#1)&)2)Sketch)the)graphs)(no)graphing)calculator))1.#f(x)#=#(–1/5)(x#+#3)(x#+#5)(x#–#2)2######## # 2.###f(x)#=#(–1/6)(x#/#3)2(x#/#1)(x#+#2)2#
##################################################################################################### # # # # # # # ##########
####
#Graphing Calculator Allowed
3. P(x) = (7x5#+#3x9#–#2x#+#4)##–##(5x2#–##2x#+#4)##
a)#Standard#Form:_________________________________________________
b) Degree ___________________ c) Classify by the # of terms: __________________________# # #
__________________4. Find p(3) for p(x) = -4x4 + 9x3 + 10x2 – 2x + 17 ##
________________________5.##Is (x + 2) a factor of p(x) = x4 + 3x3 – 3x – 10 ? #
6.###Divide#using#long)division)(2x4 – 5x3 + 7x2 + 2x + 4) ÷ (2x – 3) ))) ___________________________________ )
7.##Write a polynomial function in standard form that has zeros at 2, -1, and 3 multiplicity 2?
___________________________________
8.##Solve#0#=##x3#–#x2##–#11x#+#3##given#that#/3#is#a#zero.###
___________________________________
(Honors)#9.##Write a polynomial function in standard form that has zeros at 2, -1, and 1 ± 3i ?
___________________________________
(Honors)#10.##Find all the zeros of f(x) = x4 – 3x3 – 3x2 + 7x + 6?
___________________________________
d.) Degree________________##
e.) x/#intercepts________________##
f.) y/intercept#_____________________##
x#
y#
a.) Degree________________##
b.) x/#intercepts________________##
c.) y/intercept#_____________________##
x#
y#
Polynomial Review #2 1.##Is#/3#a#zero#of#p(x)#=##2x4#+#9x3#–#7x#+#10#?##Why#or#why#not?#
2.##Is#(x#+#7)#a#factor#of#p(x)#=#x4#+#9x3#+#15x2#+#5x#–#14#?##Why#or#why#not?#
3.#Find#p(3)#for#p(x)#=#3x4#–#11x3#–#x2#+#15x#–#12##?#
4.##Factor##p(x)#=#3x3#+#14x2#–#7x#–#10##completely,##given#p(/5)#=#0###
5.##Write#the#polynomial#in#factored#form#with#zeros:##1#multiplicity#3,##0,#/4#?#
6.##Solve#p(x)#=##x3#–#3x2#–#11x#–#7##given#that#/1#is#a#zero.#####
7.##Factor#p(x)#=#6x3#–#23x2#–#6x#+#8##if#(x#–#4)#is#a#factor.#
8.##Solve##7.#####
9.#)Sketch#the#graph#of""p(x)#=#/1(x#–#2)(x#+#3)(x#+#1)###(no#calc)#10.#Solve#p(x)#=#x3#–#11x2#+#36x#–#36##if#(x#–#6)#is#a#factor.#
11.##Solve#p(x)#=#15x3#–#119x2#–#10x#+#16##if#8#is#a#zero.#
12.##Divide#x4#−#3x3#+#18x2#−12x#+#16#by#x#–#3#using#long#division.#
13.#One#root#of#2x3#−10x2#+#9x#−4#=#0#is#4.##Find#the#other#roots.#
14.#If#3#+#2i#is#a#zero#of#a#polynomial,#what#has#to#be#another#zero?#15.#Describe#the#end#behavior#of#each:##(a)##f(x)#=#x5#−#x3#−#x2#+#x#+#2;# (b)##h(x)#=#−x4#−#9x2#16.#Approximate#to#the#nearest#tenth#the#real#zeros#of#f(x)#=#x3#−6x2#+#8x#−2.##(Use#a#calculator)#17.#For#y#=#x(x#+#3)(x#−#1)2,#determine#the#zeros#and#their#multiplicity.#
18.#Write#a#polynomial#function#with#zeros#1#and#2#(of#multiplicity#3)#in#standard#form.#
19.#Use#synthetic#division#to#find#f(−2)#if#f(x)#=#4x5#+#10x4#−#11x3#−22x2#+#20x#+#10.#20.#Factor:##2x3#+#15x2#−14x#−48##if#(x#−#2)#is#a#factor.#
21.#Determine#if#the#degree#of#the#functions#below#is#even#or#odd.##How#many#real#zeros#does#each#have?#
#a)# # # ######b)# # # # # c)# # #
# # ##
#
##
#
(Honors)#22.##Find#a#third#degree#polynomial#with#zeros#/4#and#2#–#3i.#(Honors)#23.#Write#a#cubic#equation#in#standard)form#having#zeros#3#and#2#+#i.#(Honors)#24.#Find#a#polynomial#equation#having#roots#/2#and#3#+#i.#(Honors)#25.#Find#all#zeros#for#p(x)#=#2x4#+#3x3#+#6x2#+#12x#/#8#if#2i#is#a#zero.#(no#calc)#(Honors))26.##Find#all#the#POSSIBLE#rational#roots#of#p(x)#=#3x4#+#10x3#/#8x2#+#x#/#15#(Honors)#26.#Find#all#the#roots#for:#f(x)#=#3x4#+#14x3#+#14x2#/#8x#–#8##(no#calc)#(Honors)#27.#Find#all#the#roots#for#p(x)#=#3x3#/#x2#/#6x#+#2###(no#calc)#
