Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #1 Mrs. Dailey

1) Consider the cubic function y = x3 + 2x2 – 8x.

a) Algebraically determine the zeroes of this cubic function.

b) Sketch the function on the axes given. Clearly plot and label each x-intercept.

HW#1

2) Consider the cubic function y=x3 + 2x2 -36x – 72. a) Find an appropriate y-window for the x- window shown on the axes and sketch the graph. Make the sure the window is sufficiently large to show the two turning points and all intercepts. Clearly label all x- intercepts. b) What are the solutions to the equation x3 + 2x2 -36x – 72=0? c) Based on your answers to (b), how must the expression x3 + 2x2 -36x – 72 factor?

3) Consider the quartic function y = x4 – x3 -27x2 + 25x + 50. a) Sketch the graph of this function on the axes below. Clearly mark all x-intercepts. b) Use your graph from part (a) to solve the equation x4 – x3 -27x2 + 25x + 50=0. (c) Considering your answer to (b), how does the expression x4 – x3 -27x2 + 25x + 50 factor? 4) In general, how does the number of zeroes (or x-intercepts) relate to the highest power of a polynomial? Be specific.

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #2 Mrs. Dailey 1) Create the equation of a quadratic polynomial, in standard form, that has zeroes of -5 and 2 and which passes through the point (3,-24) . Sketch the graph of the quadratic below to verify your result.

2) Create the equation of a quadratic function, in standard form, that has one zero of -3 and a turning point at (-1,-16) . Hint – try to determine the second zero of the parabola by thinking about the relationship between the first zero and the turning point (axis of symmetry). Sketch your solution below.

HW#2

3) Create the equation of a cubic whose x-intercepts are given by the set

{-6,-3,5} and which passes through the point (3,36) . Note that your leading coefficient in this case will be a non-integer. Sketch your result below.

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #3 Mrs. Dailey 1) Write each of the following ratios in simplest form.

(a) 8

2

5

20

x

x (b)

3

12

12

8

y

y

(c)

10 2

4 5

6

15

x y

x y (d)

3 7

6 10

24

12

x y

x y

2) Which of the following is equivalent to the expression

6 4

2 6

4

12

x y

x y ?

(1) 4

23

x

y (3)

3

2

3x

y

(2) 2

3

3y

x (4)

3

23

x

y

3) Simplify each of the following rational expressions.

(a) 2 25

4 20

x

x

(b) 2

2

11 24

9

x x

x

(c) 2

2

4 1

5 10

x

x x

(d) 2

2

9 4

3 4 4

x

x x

(e) 2

2

7 42

2 48

x x

x x

(f) 2

2

2 3 5

25 4

x x

x

HW#3

4) Which of the following is equivalent to the fraction 2

2

9 18

15 5

x x

x x

?

(1) 3

5

x

x

(3) 6

5

x

x

(2) 6

5

x

x

(4) 6

5

x

x

5) The rational expression 2

2

2 7 6

4

x x

x

can be equivalently rewritten as

(1) 2 3

2

x

x

(3) 2 3

2

x

x

(2) 2 1

6

x

x

(4)

3 2

2

x

x

6) Written in simplest form, the fraction 2 2

5 5

y x

x y

is equal to

(1) 5y – 5x (3) ( )

5

x y

(2) 5

y x (4)

5

x y

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #4 Mrs. Dailey

1) (Aug ‘12) Express the product of 2

2

xand 2

4 20

6 8

x

x x

in simpest form.

2) Express in simplest form: 2 2

2 8 20 5

12 2 5 3

x x

x x x x

HW#4

3) Express in simplest form:

2 2 2

2 2 2

16 9 14 3 28

2 4 2 8 16 8

x x x x x

x x x x x x

Review 1) Determine the product & simplify.

( 5 )(2 )( 4 )i i i 2) (3 - 5i)(2 + i) 3) Simplify. 7 4 ( 1 3 )i i

4) Simplify: 50 5) Simplify: 2 27

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #5 Mrs. Dailey 1) Simplify: 2) Simplify:

2 2

2 2

3 4

2 5 6

x x x

x x x x

22 6 8

2 4 20

x x x

x

3) Perform the indicated operations and express in lowest terms:

2 2 2

2 2

9 7 10 2 15

2 4 3 18 2 12

x x x x x

x x x x x

HW#5

4) Perform the indicated operations and simply completely:

3 2 2

2 4 3 2

3 6 18 2 4 2 8

4 3 16

x x x x x x

x x x x x

6) Factor: 3bc – 4ad + 6ac -2bd 7) Factor: 2a + x – 2a2x – ax2

Review 1) (5 + 2i)(3 + 4i) 2) Simplify. 5 3 (2 8 )i i 3) Determine the product & simplify. 4) Determine the product & simplify.

5 4 3 9 3 23 2i i

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #6 Mrs. Dailey

1) Express in simplest form:

2) The expression 1 1

x x

x x

is equivalent to

1 1

3x x

HW#6

3) Express in simplest form: 3 9

2 6 6 2

x

x x

4) Express in simplest form: 2

3 1 1

1 1

a

a a

5) Simplify: 2

1 1m

mn n m n n

6) Express in simplest form: 2

3 15 2

25 5

y

y y

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #7 Mrs. Dailey 1) Simplify each of the following complex fractions.

(a)

1 1

2 33 1

10 5

x

x

(b)

2

1 1

8 21 1

12 3

x

x x

2) Simplify each of the following complex fractions.

(a) 2

2

5 5

31 33

x x

x

(b)

2

33

41

28

x

x

HW#7

3) Simplify each of the following complex fractions.

(a) 2

4

4 105 10

14 40

x

x xx

x x

4) Which of the following is equivalent to 2

1 1

11

x x

x x

?

(1) 1 (3) 1

x

x

(2) 2

1x (4) 2x x

5) Factor each completely. (a) 6x2-13x + 6 (b) 8x – x2 (c) 3x2 – 7x – 20

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #8 Mrs. Dailey 1) Simplify the following complex fraction.

12

25

1

x

x

2) Simplify the following complex fraction.

1 2

10 1012 10

x

xx

HW#8

3) Simplify each of the following complex fractions.

2

2

3 2 81 4

2 125 4

xx x

x xx x

4) Factor each completely. (a) 50 – 2x2 (b) 4x2- 9 (c) 2x2 – 12x

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #9 Mrs. Dailey

1) Write each of the following rational expressions in the form r

ax b

.

(a) 6

2

x

x

(b )

10

3

x

x

(c) 2 5

2

x

x

(d)5 2

4

x

x

2) If the expression 10 11

2 1

x

x

was placed in the form 52 1

a

x

, then which of the

following would be the value of a?

(1) 6 (3) 3 (2) -7 (4) -5

HW#9

3) Use polynomial long division to simplify each of the following ratios. There should be a zero remainder.

(a)

2 5 24

3

x x

x

(b)

26 11 10

3 2

x x

x

4) Use polynomial long division to write each of the following ratios in

( )r

q xx a

, form, where q(x) is a polynomial and r is the remainder.

(a)

2 6 11

4

x x

x

(b)

2 2 25

7

x x

x

4) Use polynomial long division to write each of the following ratios in

( )r

q xx a

, form, where q(x) is a polynomial and r is the remainder.

(c)

23 17 25

4

x x

x

(d)

25 41 3

8

x x

x

5) Write each of the following in ( )r

q xx a

. The polynomial q (x) will now

be a quadratic.

(a) 3 27 17 41

5

x x x

x

(b) 3 22 11 22 25

3

x x x

x

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #10 Mrs. Dailey 1) Which of the following is the remainder when the polynomial x2 -5x+ 3 is divided by the binomial (x-8)?

(1) 107 (3) 3 (2) 27 (4) 9

2) If the ratio 22 17 42

5

x x

x

is placed in the form ( )5

rq x

x

, where q (x) is a

polynomial, then which of the following is the correct value of r?

(1) -3 (3) 18 (2) 177 (4) 7

3) When the polynomial p(x) was divided by the factor x - 7 the result was

11

7x

x

. Which of the following is the value of p(7) ?

(1) -8 (3) 11 (2) 7 (4) It does not exist

4) Which of the following binomials is a factor of the quadratic 4x2-35x+24? Try to do this without factoring but by using the Remainder Theorem.

(1) x+6 (3) x-8 (2) x-4 (4) x+2

HW#10

5) Which of the following linear expressions is a factor of the cubic polynomial x3 + 9x2 + 16x -12?

(1) x+ 6 (3) x-3 (2) x- 1 (4) x+2

6) Consider the cubic polynomial p (x)= x3 + x2 - 46x + 80.

a) Using polynomial long division, write the ratio of ( )

3

p x

x in quotient-remainder

form, ( )3

rq x

x

. Evaluate p(3) How does this help you check your quotient-

remainder form? b) Evaluate p(5). What does this tell you about the binomial x-5?

c) If ( )

( )5

p xq x

x

, then use polynomial long division to find an expression for the

polynomial.

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #11 Mrs. Dailey 1) Solve each of the following fractional equations. After “clearing” the denominators you should have a linear equation to solve.

a) 2 1 3

3 6 2

x x b)

13 4 31

2 15 6x x c)

5 13

2 2x

HW#11

2) Solve each of the fractional equations for all value(s) of x.

a) 12

8xx

b) 2

3 1 1 1

4 2 2 3x x x

3) Solve the following equation for all values of x. Express your answers in simplest a + bi form.

3

9 1

x x

x

4) Solve the following equation for all values of x. Be sure to check for extraneous roots.

11

11 11

x

x x

5) Solve each of the following equations. Be sure to check for extraneous roots

(a) 2

1 2 2

5 6 11 30

x

x x x x

b) 2

3 1 2

7 7

x

x x x x

Name ______________________ Algebra II Unit 10 Polynomial and Complex Fractions HW #12 Mrs. Dailey Solve each of the following rational inequalities. Show your answers using a number line and an appropriate notation.

1) 100

5

x

x

2) 2 1

03

x

x

3) 2

2

40

20

x

x x

4)

2

2

6 160

6

x x

x x

HW#12

5) 2

2

6 90

4 3 1

x x

x x

6)

2

2

12 360

4 4 1

x x

x x

7) 1

23

x

x

8)

2 2 4

4 3

x x

x

9) 2

1 1 3

2 2 4x x x

Name ___________________________ Date _____________ CC Algebra II Unit 10 Polynomial and Rational Functions Mrs. Dailey 1) Given a graph, state the linear factors.

1) Find x-intercepts and linear factors. Answer x-intercepts: {-2, 4,10} Linear Factors: (x+2)(x-4)(x-10)

2) Write an equation given the roots (x-intercepts) or a graph and (1) pt. 1) Rewrite roots as factors and the product of “a”. y= a (x-r1) (x-r2) (x-r3) 2) Substitute point in for “x” and “y”. 3) Solve for “a”. 4) Write equation using “a” and factors. 5) y=a(x- r)(x-r)(x-r)

3) Facts about Odd Functions (greatest exponent is odd): y = x3+2x2 -4

1) y = axb “b” is odd 2) (x,y) (-x,-y) (negated y)

If (3,7) (-3,-7) lies on graph 3) Graphically End Behavior: 1 up and 1 down

4) Facts about Even Functions (Greatest exponent is Even): y = x6+2x2 -4

1) y = axb “b” is even (“a” dictates how it opens “a” positive Opens up, “a” negative opens down.)

2) (x,y) (-x,y) (same y) If (3,7) (-3,7) lies on graph

3) Graphically End Behavior: Same Direction (Both up or both down)

4) Sign of Leading Coefficient dictates open up (positive) and opens down (negative).

Unit 10 Review Tips

1 MC 1 Pt2

1 MC

2 MC 1 Pt 2

5) Domain of Fractions: Can be anything but values that make denominator =0. So, set denominator =0 to find what is not allowed in domain 6) Factoring by Grouping 7) Use Remainder Theorem to check to see if a binomial is a factor.

8) Add/Subtract Fractions 1) Find LCD and change fractions 2) Keep denominator and combine numerator

9) Multiply/Divide Fractions 1) Factor Numerator and denominator 2) Cancel. 3) Multiply what is left.

10) Absolute Value Inequalities 1) Solve inequality as an equation (Open/Closed circle based on inequality) 2) Set denominator =0 (Critical values always open circle) 3) Plot critical values on # line 4) Test values to find out where to shade

11) Long Division 12) Solve Rational Equations

Which of the following is a factor of the cubic polynomial x3-10x2+11x+70?? (1) x+10 (2) x- 2 (3) x-7 (4) x+ 5

Remainder Theorem: Remainder must be 0 if binomial is a factor.

So…. Check to see P(-10) = 0 Check to see P(2) =0 Check to see P(7) = 0 Check to see P(-5) = 0

1) Look for two numbers that Multiply: 48 and Add: -19 22 19 24x x

-16 and -3 2) Rewrite as 4 terms

22 16 3 24x x x 3) Factor 1st two terms and then factor 2nd two terms

2 ( 8) 3( 8)

2 3 8

x x x

x x

1 Pt 2

1 MC 1 Pt 2

2 MC

1 MC 2 Pt 2

1 MC

1 MC 1Pt 2

1Pt 2

Name _______________________________ Date _____________ CC Algebra II Unit 10 Polynomial and Rational Functions Mrs. Dailey 1) Which of the following could be the equation of the polynomial graph shown

below? (1) 5 3y x x x

(2) 3 5y x x x

(3) 2 5 3y x x x

(4) 2 3 5y x x x

` 2) The cubic If polynomial shown graphed below passes through the point (6, 192) as shown. Which of the following is the value of the leading coefficient of

the polynomial? (1) -7

(2) -3

(3) 1

3

(4) 5

When graph only touches x-axis …and does not pass through x-

axis…a double root occurs.

1) Find x-intercept to write linear factors. 2) y = a(x-r1)(x-r2)(x-r3) 3) Plug in (x,y) from pt to find value of a.

Review

3) Which of the following equations represents an identity?

(1) 2 24 16x x (3) 2 9 3 3 3x x x x

(2) 5 2 1 10 6x x (4) 24 2 2x x x

4) Which of the following values of x is not in the domain of the function

2

9

5 14

xy

x x

?

(1) x = 9 (3) x = -14 (2) x = 2 (4) x= -2

5) The rational expression

2

2

4 21

49

x x

x

can be simplified to which of the

following, assuming x≠ 7?

(1) 3

7 (3) 4 21

49

x

(2) 7

7

x

x

(4) 3

7

x

x

6) As long as x and y are not equal to zero, the expression 232 5

4 4

168

3 9

x yx y

x y is

equivalent to

(1) 4 62

5x y (3)

8 73

2x y

(2) 6 318x y (4)

2 520

3x y

Identities are equations that are true for all values of x. In terms of manipulations, they must "look" the same on both sides of the equality.

Fractions Not allowed to have zero in the denominator. Do find the domain, set denominator = 0.

Factor numerator. Factor denominator. Cancel.

Flip 2nd fraction and multiply. Then use laws of exponents to reduce.

7) Which of the following represents the sum 1 1

x y expressed as a single

fraction?

(1) 2

x y (3)

1

x y

(2) x y

xy

(4)

xy

x y

8) Which of the following is equivalent to the difference 2

2 2

2 8

16 16

x x

x x

?

(1) 2

4

x

x

(3) 6

2

x

x

(2) 4

8

x

x

(4) 3

4

x

x

9) For x≠0, the complex fraction

11

11

x

x

simplifies to

(1) 1 (3) 1

1

x

x

(2) 2 1x

x

(4) 1x

x

Get common denominator and add fractions

Get common denominator and subtract fractions

10) The complex fraction

1 1

x y xy

can be rewritten as

(1) 2

1

x xy

(3) 2 2

y

x y

(2) 2

1

y (4) 2

1 y

y xy

11) The quotient 10

2

x

x

can be written as

(1) -5 (3) 1 – 5x

(2) 121

2x

(4) 8

12x

12) If 6 1

3 7

x

x

was written in the form 2

3 7

b

x

, then b =

(1) -1 (3) 12 (2) 7 (4) -15 13) Which of the following is a factor of the cubic polynomial x3-10x2+11x+70?? (1) x+10 (3) x-7 (2) x- 2 (4) x+ 5

Long Division

Long Division

Remainder Theorem: Remainder = 0 for binomial to be a factor.

14) The smallest solution to the equation 2 30

2 4

x

x x

is which of the following?

(1) x = -4 (3) x = 5 (2) x = 6 (4) x = 10 15) What is the remainder when x2 + 8x - 7 is divided by x- 2? (1) 13 (3) -19 (2) -27 (4) 5 16) Which of the following represents the solution set of the equation:

2

16 61

x x ?

(1) 4, 6 (3) 4, 4

(2) 8, 2 (4) 2, 8

Cross-Multiply and solve.

Long Division

17) The solution set to the inequality 2 9

01

x

x

is which of the following?

(1) 121 4x (3) 1

21 or 4x x

(2) 121 4x (4) 1

21 or 4x x 18) Given the cubic polynomial f(x) = x3 – 5x2 -4x + 20 answer the following. (a) Find the x-intercepts of this function algebraically. Show how you arrived at

your answer. (b) Explain why the graph below could not represent that of f(x).

Find critical values. 1) Solve as equation. 2) Set denominator =0 3) Plot on # line and test

Factor by grouping.

19) The cubic polynomial below has zeroes at x =-4 and x=6 only and passes through the point (2,36) as shown. Algebraically determine its equation in factored form. Show how you arrived at your answer. 20) Determine any value(s) of x that do not lie in the domain of the function

2

2

2 15

2 19 24

x xf x

x x

. Explain how you arrived at your value(s).

1) Write roots as factors y= a(x-r1)(x-r2)(x-r3) 2) Find “a” 3) Substitute a back in to 1.

Cannot have a denominator = 0 1) Set denominator = 0 These values are not in domain

21) Express the following division problem in simplest terms.

2 2

2 2

5 30 36

10 4 12

x x x

x x x

22) Write the following complex fraction in simplest form.

102

55

xx

x

1) Flip 2nd fraction and change to multiplication. 2) Factor all and cancel

23) The rational expression 3 24 2 8 10

5

x x x

x

can be written as 5

kp x

x

,

where p(x) is a quadratic polynomial and k is a constant. (a) Determine the equation for p(x). Show how you arrived at your answer. (b) Is x - 5 a factor of 4x3 – 2x2 + 8x + 10? Explain how your reasoning. 24) For the cubic polynomial f(x) = x3 – x2 – 44x - 96 answer the following. (a) Given that f(-3) = 0, what binomial must be a factor of this polynomial. (b) Given your answer to (a), algebraically determine the two other factors of

f(x).

Use Long Division.

25) Solve the following equation for all value(s) of x:

2

5 1 4

2 2

x x

x x x x

26) Algebraically determine the solution set to the inequality shown below. Graph your solution on the number line given.

5 1

33

x

x

27) Solve the inequality shown below and graph its solution set on the number line provided.

2 5 500

3

x x

x

\ Name ___________________________ Date _____________ CC Algebra II Unit 10 Polynomial and Rational Functions Mrs. Dailey 1) Given a graph, state the linear factors.

2) Find x-intercepts and linear factors. Answer x-intercepts: {-2, 4,10} Linear Factors: (x+2)(x-4)(x-10)

2) Write an equation given the roots (x-intercepts) or a graph and (1) pt. 1) Rewrite roots as factors and the product of “a”. y= a (x-r1) (x-r2) (x-r3) 2) Substitute point in for “x” and “y”. 3) Solve for “a”. 4) Write equation using “a” and factors. 5) y=a(x- r)(x-r)(x-r)

3) Facts about Odd Functions (greatest exponent is odd): y = x3+2x2 -4

4) y = axb “b” is odd 5) (x,y) (-x,-y) (negated y)

If (3,7) (-3,-7) lies on graph 6) Graphically End Behavior: 1 up and 1 down

4) Facts about Even Functions (Greatest exponent is Even): y = x6+2x2 -4

5) y = axb “b” is even (“a” dictates how it opens “a” positive Opens up, “a” negative opens down.)

6) (x,y) (-x,y) (same y) If (3,7) (-3,7) lies on graph

7) Graphically End Behavior: Same Direction (Both up or both down)

8) Sign of Leading Coefficient dictates open up (positive) and opens down (negative).

Unit 10 Review Tips

5) Domain of Fractions: Can be anything but values that make denominator =0. So, set denominator =0 to find what is not allowed in domain 6) Factoring by Grouping 7) Use Remainder Theorem to check to see if a binomial is a factor.

8) Add/Subtract Fractions 3) Find LCD and change fractions 4) Keep denominator and combine numerator

9) Multiply/Divide Fractions 4) Factor Numerator and denominator 5) Cancel. 6) Multiply what is left.

10) Absolute Value Inequalities 6) Solve inequality as an equation (Open/Closed circle based on inequality) 7) Set denominator =0 (Critical values always open circle) 8) Plot critical values on # line 9) Test values to find out where to shade

11) Long Division 12) Solve Rational Equations

Which of the following is a factor of the cubic polynomial x3-10x2+11x+70?? (1) x+10 (2) x- 2 (3) x-7 (4) x+ 5

Remainder Theorem: Remainder must be 0 if binomial is a factor.

So…. Check to see P(-10) = 0 Check to see P(2) =0 Check to see P(7) = 0 Check to see P(-5) = 0

1) Look for two numbers that Multiply: 48 and Add: -19 22 19 24x x

-16 and -3 2) Rewrite as 4 terms

22 16 3 24x x x 3) Factor 1st two terms and then factor 2nd two terms

2 ( 8) 3( 8)

2 3 8

x x x

x x

Name _______________________________ Date _____________ CC Algebra II Unit 10 Polynomial and Rational Functions Mrs. Dailey 1) Which of the following could be the equation of the polynomial graph shown

below? (1) 5 3y x x x

(2) 3 5y x x x

(3) 2 5 3y x x x

(4) 2 3 5y x x x

` 2) The cubic If polynomial shown graphed below passes through the point (6, 192) as shown. Which of the following is the value of the leading coefficient of

the polynomial? (1) -7

(2) -3

(3) 1

3

(4) 5

When graph only touches x-axis …and does not pass through x-

axis…a double root occurs.

1) Find x-intercept to write linear factors. 2) y = a(x-r1)(x-r2)(x-r3) 3) Plug in (x,y) from pt to find value of a.

Review

3) Which of the following equations represents an identity?

(1) 2 24 16x x (3) 2 9 3 3 3x x x x

(2) 5 2 1 10 6x x (4) 24 2 2x x x

4) Which of the following values of x is not in the domain of the function

2

9

5 14

xy

x x

?

(1) x = 9 (3) x = -14 (2) x = 2 (4) x= -2

5) The rational expression

2

2

4 21

49

x x

x

can be simplified to which of the

following, assuming x≠ 7?

(1) 3

7 (3) 4 21

49

x

(2) 7

7

x

x

(4) 3

7

x

x

6) As long as x and y are not equal to zero, the expression 232 5

4 4

168

3 9

x yx y

x y is

equivalent to

(1) 4 62

5x y (3)

8 73

2x y

(2) 6 318x y (4)

2 520

3x y

Identities are equations that are true for all values of x. In terms of manipulations, they must "look" the same on both sides of the equality.

Fractions Not allowed to have zero in the denominator. Do find the domain, set denominator = 0.

Factor numerator. Factor denominator. Cancel.

Flip 2nd fraction and multiply. Then use laws of exponents to reduce.

7) Which of the following represents the sum 1 1

x y expressed as a single

fraction?

(1) 2

x y (3)

1

x y

(2) x y

xy

(4)

xy

x y

8) Which of the following is equivalent to the difference 2

2 2

2 8

16 16

x x

x x

?

(1) 2

4

x

x

(3) 6

2

x

x

(2) 4

8

x

x

(4) 3

4

x

x

9) For x≠0, the complex fraction

11

11

x

x

simplifies to

(1) 1 (3) 1

1

x

x

(2) 2 1x

x

(4) 1x

x

Get common denominator and add fractions

Get common denominator and subtract fractions

10) The complex fraction

1 1

x y xy

can be rewritten as

(1) 2

1

x xy

(3) 2 2

y

x y

(2) 2

1

y (4) 2

1 y

y xy

11) The quotient 10

2

x

x

can be written as

(1) -5 (3) 1 – 5x

(2) 121

2x

(4) 8

12x

12) If 6 1

3 7

x

x

was written in the form 2

3 7

b

x

, then b =

(1) -1 (3) 12 (2) 7 (4) -15 13) Which of the following is a factor of the cubic polynomial x3-10x2+11x+70?? (1) x+10 (3) x-7 (2) x- 2 (4) x+ 5

Long Division

Long Division

Remainder Theorem: Remainder = 0 for binomial to be a factor.

14) The smallest solution to the equation 2 30

2 4

x

x x

is which of the following?

(1) x = -4 (3) x = 5 (2) x = 6 (4) x = 10 15) What is the remainder when x2 + 8x - 7 is divided by x- 2? (1) 13 (3) -19 (2) -27 (4) 5 16) Which of the following represents the solution set of the equation:

2

16 61

x x ?

(1) 4, 6 (3) 4, 4

(2) 8, 2 (4) 2, 8

Cross-Multiply and solve.

Long Division

17) The solution set to the inequality 2 9

01

x

x

is which of the following?

(1) 121 4x (3) 1

21 or 4x x

(2) 121 4x (4) 1

21 or 4x x 18) Given the cubic polynomial f(x) = x3 – 5x2 -4x + 20 answer the following. (a) Find the x-intercepts of this function algebraically. Show how you arrived at

your answer. (b) Explain why the graph below could not represent that of f(x).

Find critical values. 1) Solve as equation. 2) Set denominator =0 3) Plot on # line and test

Factor by grouping.

19) The cubic polynomial below has zeroes at x =-4 and x=6 only and passes through the point (2,36) as shown. Algebraically determine its equation in factored form. Show how you arrived at your answer. 20) Determine any value(s) of x that do not lie in the domain of the function

2

2

2 15

2 19 24

x xf x

x x

. Explain how you arrived at your value(s).

1) Write roots as factors y= a(x-r1)(x-r2)(x-r3) 2) Find “a” 3) Substitute a back in to 1.

Cannot have a denominator = 0 1) Set denominator = 0 These values are not in domain

21) Express the following division problem in simplest terms.

2 2

2 2

5 30 36

10 4 12

x x x

x x x

22) Write the following complex fraction in simplest form.

102

55

xx

x

1) Flip 2nd fraction and change to multiplication. 2) Factor all and cancel

23) The rational expression 3 24 2 8 10

5

x x x

x

can be written as 5

kp x

x

,

where p(x) is a quadratic polynomial and k is a constant. (a) Determine the equation for p(x). Show how you arrived at your answer. (b) Is x - 5 a factor of 4x3 – 2x2 + 8x + 10? Explain how your reasoning. 24) For the cubic polynomial f(x) = x3 – x2 – 44x - 96 answer the following. (a) Given that f(-3) = 0, what binomial must be a factor of this polynomial. (b) Given your answer to (a), algebraically determine the two other factors of

f(x).

Use Long Division.

25) Solve the following equation for all value(s) of x:

2

5 1 4

2 2

x x

x x x x

26) Algebraically determine the solution set to the inequality shown below. Graph your solution on the number line given.

5 1

33

x

x

27) Solve the inequality shown below and graph its solution set on the number line provided.

2 5 500

3

x x

x