Polynomials Notebook.notebook 1 October 02, 2012 May 1112:48 PM Unit 2 Polynomials At a birthday party, Crusty the Clown took the age of the birthday girl and multiplied it by 6, subtracted 2, multiplied it by 7, and then added 3. The resulting number was 493. How old is the birthday girl? May 1112:48 PM POLYNOMIALS May 1112:58 PM A polynomial is made up of terms that are only added, subtracted or multiplied. What is a polynomial??? May 147:59 AM A polynomial with 3 terms is called a trinomial . Monomial = polynomial with 1 term Binomial = polynomial with 2 terms There is also a quadrinomial (4 terms) & a quintinomial (5 terms) but we rarely use these words. May 147:53 AM The degree of a polynomial with only one variable is the largest exponent of that variable. If the polynomial has more than one variable, you need to look at each term. You find the degree of the polynomial by adding the exponents of each term and naming the largest power . The degree of this trinomial is 4. The degree of this trinomial is 3. May 148:13 AM Find the degree of each polynomial. 1. 3x 2 2. 4x 2 y 3 ‐ 1 3. 3x 2 +4x 2 y 3 ‐ 5z 4
Transcript
Polynomials Notebook.notebook
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October 02, 2012
May 1112:48 PM
Unit 2 PolynomialsAt a birthday party, Crusty the Clown took the age of the birthday girl and multiplied it by 6,
subtracted 2, multiplied it by 7, and then added 3. The resulting number was 493. How old is
the birthday girl?
May 1112:48 PM
POLYNOMIALS
May 1112:58 PM
A polynomial is made up of terms that are only added, subtracted or multiplied.
What is a polynomial???
May 147:59 AM
A polynomial with 3 terms is called a trinomial.
Monomial = polynomial with 1 termBinomial = polynomial with 2 terms
There is also a quadrinomial (4 terms)& a quintinomial (5 terms) but we rarely
use these words.
May 147:53 AM
The degree of a polynomial with only one variable is the largest exponent of that variable.
If the polynomial has more than one variable, you need to look at each term. You find the degree of the polynomial by adding the exponents of each term and naming the largest power.
The degree of this trinomial is 4.
The degree of this trinomial is 3.
May 148:13 AM
Find the degree of each polynomial. 1. 3x2
2. 4x 2y3 ‐ 1
3. 3x2 +4x2y3 ‐ 5z4
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October 02, 2012
May 147:55 AM
The standard form of a polynomial is to put the terms with the highest degree first.
Arrange the following in standard form:
1. 4x2 ‐ 2x + 3 ‐ 5x3
2. ‐1 + x3 ‐ 9x2 + 8x
May 149:58 AM
Quiz Time!!!
Just kidding...but here are a few questions...
May 148:32 AM May 148:34 AM
May 1410:43 AM
Adding & Subtracting Polynomials
May 1410:48 AM
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October 02, 2012
May 1410:48 AM May 1411:12 AM
May 1410:56 AM May 1411:11 AM
May 1410:49 AM May 1411:14 AM
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October 02, 2012
May 1411:14 AM May 1411:15 AM
May 1410:40 AM
Remember, to add polynomials simply combine like terms. To subtract polynomials, distribute
the -1 and then combine like terms.
May 1410:39 AM
May 1410:39 AM May 1411:23 AM
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October 02, 2012
May 1411:30 AM
Homework:p. 113 #2- 58 eoe
May 1411:31 AM
10. 22.
30. 46.
Sep 287:39 AM
Answers to p. 113 #2 58 eoe
2. 4 6. 3
10. 1 14. 10t + 8
18. 5a b 5ab 22. 5x 6
26. 2x 2x 2 30. 3x x x 6
34. 11x + 8 38. x + 4x + x 5
42. 9x 9x 7 46. n n + 1
50. 3x + 7 54. 3x 2x + 10
58. 4x 4
May 1112:56 PM
Maya takes a number, multiplies it by 4, subtracts 5, multiplies by 4 again
and then adds 8. The resulting number is 228. What number did
Maya start with?
WARM UP
May 1112:56 PM
Multiplying Monomials
May 1411:31 AM
Examples:1. 5(-7x) =
2. -5y(-7x2) =
3. 5x3y2(7xy4) =
4. 5xy(-9x2y3)(4x3y) =
5. 3x(-2x2y3)(-5x3y) =
To multiply 2 polynomials:• Multiply coefficients• Add exponents of like bases
Remember that whenyou multiply the samebases, you ADD the exponent. Don't forgetto multiply coefficients!
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October 02, 2012
Jul 129:14 AM
Can you put together the "old" stuff you learned yesterday with the "new" stuff from today???
Simplify:
1. (x - 2y) - 4(-x + y) + 2(-4x + 5y)
2. 2m4n(7m5n3 + 6mn2) - 6mn2(2m4n - 7m5n3)
May 1412:51 PM
Power to a Power • Distribute the outside exponent
to all inside exponentsOR
• Rewrite what is inside the parenthesis the same # as the
outside exponent and follow your rules for multiplying polynomials.
May 1412:51 PM
Examples:1. (3xy2)3 =
2. (-x4y5)4 =
3. (-5a2b2c)3 =
May 1412:57 PM
Homework:p. 120 #6- 78Multiples of 6!
Sep 282:49 PM May 1412:57 PM
Answers to p. 120 #6 78 (multiples of 6)
6. 12.
18. 24.
30. 36.
42. 48.
54. 60.
66. 72.
76.
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October 02, 2012
May 141:01 PM
Aidan spent $14.95 on ingredients for cookies that will be sold at a bake sale. If each cookie will sell for $0.40, how many cookies must Aidan sell to earn a
profit?
Warm Up
May 141:01 PM
Multiplying Polynomials
May 1112:56 PM May 141:10 PM
May 141:10 PM May 141:11 PM
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October 02, 2012
May 141:11 PM May 141:12 PM
Multiplying 2 Binomials
See next page for the process and answer...
May 141:13 PM May 141:13 PM
May 141:14 PM
A square has a side represented by x -2. Find the area of the square in terms of x.
May 141:14 PM
A rectangle has a length represented by 5x + 3 and a width represented by 2x - 7. Find the area and write your answer in standard form.
Polynomials Notebook.notebook
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October 02, 2012
May 141:14 PM
A checker board is made up of 16 congruent squares. Each square has a length represented by x - 7. Find the area of 1 square in terms of x. Then find the area of the entire checker board in terms of x.
May 141:14 PM
May 141:15 PM May 141:16 PM
Homework:p. 127
#2- 58 eoe
May 141:16 PM May 1112:57 PM
Quiz Today
CR #2 due tomorrow!
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October 02, 2012
May 1112:56 PM
A hot air balloon begins ascending at the rate of 18 feet per minute. A second hot air balloon begins ascending 8 minutes later at a rate of 27 feet per minute. How many minutes will it take for the second hot air balloon to catch the first hot air
balloon?
Warm Up
May 1112:56 PM
Factoring
May 141:55 PM
The numbers that are multiplied together are called
factors of the final number.
What is a factor?
May 157:52 AM
May 157:58 AM
Greatest Common Factor
May 158:02 AM
Try factoring these examples:1. 28y2 - 4y =
2. 7x3 + 10x2 =
3. 12x3y4 - 39 x4y3 =
4. 5x + 7x2 + 9x4 =
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October 02, 2012
May 158:02 AM
Factoring the Difference of 2 Perfect Squares
May 158:10 AM
May 158:12 AM May 158:13 AM
Try factoring these examples:
1. x2 - 100 =
2. 9 - 25y2 =
3. 16x2 - 1 =
4. x6 - y10 =
5. 49x2 - 64y2 =
May 158:13 AM May 141:55 PM
Homework:p. 135 #28- 40 even
p. 142 #1- 13 odd
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October 02, 2012
May 158:18 AM May 1112:56 PM
Warm UpThe formula 180﴾n 2﴿ is used to find the
sum of the interior angles of any polygon. If n represents the number of sides the polygon has, find the total sum of the interior angles of an octagon.
May 1112:56 PM
Factoring Trinomials
May 158:20 AM
May 158:28 AM
= (x )(x )
= (x 3)(x 4)
= (x + 3)(x + 4) + for the letter t for the word "BOTH." Both signs are the same as the middle
sign.
Positive. Therefore both
signs are positive.
Here's a little trick (Bigger/ Both)
to help you figure out the signs.
What 2 #s multiply to give you 12 but ADD to
give you 7?
May 158:28 AM
= (x )(x )
= (x 9)(x 5)
= (x - 9)(x + 5) - for the letter i for the word "BIGGER." The bigger # gets the
same sign as the middle sign.
Negative. Therefore the
bigger # (9) gets the - and the smaller # (5) gets the +.
Here's a little trick (Bigger/ Both)
to help you figure out the signs.
What 2 #s multiply to give you 45 but SUBTRACT to
give you 4?
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October 02, 2012
May 158:25 AM May 1510:05 AM
May 1510:07 AM May 158:27 AM
May 1510:10 AM May 158:18 AM
Homework:Worksheet
Polynomials Notebook.notebook
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October 02, 2012
May 158:18 AM May 1112:57 PM
Warm UpAkira scored 18 points in her last
basketball game, p of which came in the 4th quarter of the game. Write an
expression that can be used to find the number of points she scored in the first 3
quarters.
May 1112:57 PM
Factoring by Grouping
May 1510:39 AM
Let's start out simple...Factor the following:
1.
2.
Factoring by
grouping will pop
up if you go on to
take Calculus!
May 1510:44 AM
KEY POINT!!!
May 1510:56 AM
Polynomials Notebook.notebook
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October 02, 2012
May 1510:57 AM May 1511:15 AM
The Arc Method of Factoring
May 1511:23 AM
Factor:5x2 8x + 3
Wait there are no 2 #'s that multiply to give me 3 and add to give me 8!!!(3)(1) = 33 + 1 = 4
What am I supposed to do????
We will use the Arc Method of Factoring!
May 1511:17 AM
The Arc Method of FactoringSteps:1. Take the coefficient in front of the x2 and multiply it by the last term.2. Replace the last term with this new number.3. Factor this new trinomial. 4. "Un arc" the factors by dividing the #'s by the original # in from of the x2.5. Reduce all fractions.6. If the fraction is not divisible, move the denominator in from of your x term.
1. Factor a GCF from the expression, if possible. 2. Factor a Trinomial, if possible. 3. Factor a Difference Between Two Squares as many times as possible.
x3 + 7x2 + 12x = x﴾x2 + 7x + 12﴿ x﴾x + 3﴿﴾x + 4﴿
To factor x 3 + 7x2 + 12x you need to FACTOR COMPLETELY
May 151:55 PM
Factor each of the following COMPLETELY:
1. 2x2 8
2. n3 49n
3. 25x2 100
May 151:55 PM
Factor each of the following COMPLETELY:
1. 18x3 + 39x2 15x
2. 3x2 + 15x 18
May 168:11 AM
Homework:Worksheet
Quiz tomorrow!!!
May 168:11 AM
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October 02, 2012
May 168:13 AM
Warm Up
What is the next number in the
sequence?
1, 2, 4, 8, 16, ...
May 168:13 AM
Solving Quadratic Equations
May 1112:58 PM
What is a quadratic? A quadratic equation???
May 168:17 AM
May 168:22 AM
It must get harder than this, right???
May 168:19 AM
What do we call the highest power of a
polynomial????
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October 02, 2012
May 168:25 AM
Solve the equation below for x. How many values will you end up with? Why?
