Factors Greatest Common Factorscontent.njctl.org/courses/math/archived-coursesunits-prior-to-comm… · The first step in factoring is to determine its greatest monomial factor. If

Polynomials

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Table of Contents

· Factors and GCF· Factoring out GCF's

· Factoring 4 Term Polynomials

· Factoring Trinomials x2 + bx + c· Factoring Using Special Patterns· Factoring Trinomials ax2 + bx + c

· Mixed Factoring· Solving Equations by Factoring

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Factorsand

Greatest Common Factors

Return toTable ofContents

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10

11121314151617181920

Number BankFactors of 10 Factors of 15

FactorsUniqueto 15

FactorsUnique to 10

Factors 10 and 15have in common

What is the greatest common factor (GCF) of 10 and 15?

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10

11121314151617181920

Number BankFactors of 12 Factors of 18

FactorsUniqueto 18

FactorsUnique to 12

Factors 12 and 18have in common

What is the greatest common factor (GCF) of 12 and 18?

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1 What is the GCF of 12 and 15?

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5 What is the GCF of 28, 56 and 42?

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Variables also have a GCF.

The GCF of variables is the variable(s) that is in each term raised to the lowest exponent given.

Example: Find the GCF

and

and and and

andand

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6 What is the GCF of

A

B

C

D

and ?

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Factoringout GCFs


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The first step in factoring is to determine its greatest monomial factor. If there is a greatest monomial factor other than 1, use the distributive property to rewrite the given polynomial as the product of this greatest monomial factor and a polynomial.

Example 1 Factor each polynomial.

a) 6x4 - 15x3 + 3x2

3x2 (2x2 - 5x + 1)

GCF: 3x2

Reduce each term of the polynomial dividingby the GCF

Find the GCF 3x2

3x2 3x2 3x2

3x26x4 15x3

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The first step in factoring is to determine its greatest monomial factor. If there is a greatest monomial factor other than 1, use the distributive property to rewrite the given polynomial as the product of this greatest monomial factor and a polynomial.

Example 1 Factor each polynomial.

b) 4m3n - 7m2n2

m2n(4n - 7n)

GCF: m2nReduce each term of the polynomial dividingby the GCF

Find the GCF

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Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor.Example 2 Factor each polynomial.a) y(y - 3) + 7(y - 3)

(y - 3)(y + 7)

GCF: y - 3Reduce each term of the polynomial dividingby the GCF

Find the GCF(y - 3) y(y - 3)

(y - 3)7(y - 3)(y - 3)

+( (

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Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor.Example 2 Factor each polynomial.b)

GCF:

Reduce each term of the polynomial dividingby the GCF

Find the GCF

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In working with common binomial factors, look for factors that are opposites of each other. For example: (x - y) = - (y - x) because

x - y = x + (-y) = -y + x = -(y - x)

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10 True or False: y - 7 = -1( 7 + y)

True

False

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11 True or False: 8 - d = -1( d + 8)

True

False

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12 True or False: 8c - h = -1( -8c + h)

True

False

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13 True or False: -a - b = -1( a + b)

True

False

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14 If possible, Factor

A

B

C

D Already Simplified

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A

B

C


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A

B

C


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A

B

C


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A

B

C


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A

B

C


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Factoring Trinomials:x2 + bx + c


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A polynomial that can be simplified to the form ax + bx + c , where a ≠ 0, is called a quadratic polynomial.Quadratic term.

Linear term.

Constant term.

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A quadratic polynomial in which b ≠ 0 and c ≠ 0 is called aquadratic trinomial. If only b=0 or c=0 it is called a quadratic binomial. If both b=0 and c=0 it is a quadratic monomial.Examples: Choose all of the description that apply.

Quadratic

Linear

Constant

Trinomial

Binomial

Monomial

Cubic

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20 Choose all of the descriptions that apply to:

A QuadraticB LinearC ConstantD Trinomial

E BinomialF Monomial

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Simplify.

1) (x + 2)(x + 3) = _________________________2) (x - 4)(x - 1) = _________________________3) (x + 1)(x - 5) = ________________________4) (x + 6)(x - 2) = ________________________

RECALL … What did we do?? Look for a pattern!!

x2 - 5x + 4x2 - 4x - 5

x2 + 4x - 12

Slide each polynomial from the circle to the correct expression.

x2 + 5x + 6

AnswerBank

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(x - 8)(x - 1)

Examples:

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24 The factors of 12 will have what kind of signs given the following equation?

A Both positive

B Both Negative

C Bigger factor positive, the other negative

D The bigger factor negative, the other positive

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25 The factors of 12 will have what kind of signs given the following equation?

A Both positive

B Both negative



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26 Factor

A (x + 12)(x + 1)

B (x + 6)(x + 2)

C (x + 4)(x + 3)

D (x - 12)(x - 1)

E (x - 6)(x - 1)

F (x - 4)(x - 3)

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27 Factor

A (x + 12)(x + 1)

B (x + 6)(x + 2)

C (x + 4)(x + 3)

D (x - 12)(x - 1)

E (x - 6)(x - 1)

F (x - 4)(x - 3)

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28 Factor

A (x + 12)(x + 1)

B (x + 6)(x + 2)

C (x + 4)(x + 3)

D (x - 12)(x - 1)

E (x - 6)(x - 1)

F (x - 4)(x - 3)

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29 Factor

A (x + 12)(x + 1)

B (x + 6)(x + 2)

C (x + 4)(x + 3)

D (x - 12)(x - 1)

E (x - 6)(x - 1)

F (x - 4)(x - 3)

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Examples

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30 The factors of -12 will have what kind of signs given the following equation?

A Both positive

B Both negative



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31 The factors of -12 will have what kind of signs given the following equation?

A Both positive

B Both negative



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32 Factor x2 - 4x - 12

A (x + 12)(x - 1)

B (x + 6)(x - 2)

C (x + 4)(x - 3)

D (x - 12)(x + 1)E (x - 6)(x + 2)

F (x - 4)(x + 3)

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33 Factor

A (x + 12)(x - 1)

B (x + 6)(x - 2)

C (x + 4)(x - 3)

D (x - 12)(x + 1)

E (x - 6)(x + 1)

F (x - 4)(x + 3)

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34 Factor

A (x + 12)(x - 1)

B (x + 6)(x - 2)

C (x + 4)(x - 3)

D (x - 12)(x + 1)

E (x - 6)(x + 1)

F (x - 4)(x + 3)

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Mixed Practice

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36 Factor the following

A (x - 2)(x - 4)

B (x + 2)(x + 4)

C (x - 2)(x +4)

D (x + 2)(x - 4)

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A (x - 3)(x - 5)

B (x + 3)(x + 5)

C (x - 3)(x +5)

D (x + 3)(x - 5)

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A (x - 3)(x - 4)

B (x + 3)(x + 4)

C (x +2)(x +6)

D (x + 1)(x+12)

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A (x - 2)(x - 5)

B (x + 2)(x + 5)

C (x - 2)(x +5)

D (x + 2)(x - 5)

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Factor out

Factor:

STEP4

STEP3

STEP2

STEP1

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40 Factor completely:

A

B

C

D

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A

B

C

D

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A

B

C

D

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A

B

C

D

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A

B

C

D

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Factoring UsingSpecial Patterns


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When we were multiplying polynomials we had special patterns.

Square of SumsDifference of SumsProduct of a Sum and a Difference

If we learn to recognize these squares and products we can use them to help us factor.

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Perfect Square TrinomialsThe Square of a Sum and the Square of a difference have products that are called Perfect Square Trinomials.

How to Recognize a Perfect Square Trinomial:Recall:

Observe the trinomial· The first term is a perfect square.· The second term is 2 times square root of the first term times the square root of the third. The sign is plus/minus.· The third term is a perfect square.

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Examples of Perfect Square Trinomials

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Is the trinomial a perfect square?Drag the Perfect Square Trinomials into the Box.

Only Perfect Square Trinomials will remain visible.

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45 Factor

A

B

C

D Not a perfect Square Trinomial

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46 Factor

A

B

C


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47 Factor

A

B

C


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Examples of Difference of Squares

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48 Factor

A

B

C

D Not a Difference of Squares

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49 Factor

A

B

C


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50 Factor

A

B

C


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51 Factor using Difference of Squares:

A

B

C


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Factoring Trinomials:ax2 + bx + c


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How to factor a trinomial of the form ax² + bx + c.Example: Factor 2d² + 15d + 18

Find the product of a and c : 2 ∙ 18 = 36

Now find two integers whose product is 36 and whose sum is equal to “b” or 15.

Now substitute 12 + 3 into the equation for 15. 2d² + (12 + 3)d + 18 Distribute 2d² + 12d + 3d + 18 Group and factor GCF 2d(d + 6) + 3(d + 6) Factor common binomial (d + 6)(2d + 3)Remember to check using FOIL!

1, 362, 183, 121 + 36 = 372 + 18 = 203 + 12 = 15

Factors of 36 Sum = 15?

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Factor. 15x² - 13x + 2a = 15 and c = 2, but b = -13Since both a and c are positive, and b is negative we needto find two negative factors of 30 that add up to -13

Factors of 30 Sum = -13?-1, -30-2, -15-3, -10-5, -6

-1 + -30 = -31-2 + -15 = -17-3 + -10 = -13-5 + -6 = -11

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Factor 6y² - 13y - 5

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A polynomial that cannot be written as a product of two polynomials is called a prime polynomial.

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53 Factor

A

B

C

D Prime Polynomial

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54 Factor

A

B

C

D Prime Polynomial

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55 Factor

A

B

C

D Prime Polynomial

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Factoring 4 TermPolynomials


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Polynomials with four terms like ab - 4b + 6a - 24, can befactored by grouping terms of the polynomials.

Example 1: ab - 4b + 6a - 24(ab - 4b) + (6a - 24) Group terms into binomials that can be factored using the distributive property b(a - 4) + 6(a - 4) Factor the GCF(a - 4) (b + 6) Notice that a - 4 is a common binomial factor and factor!

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Example 2:6xy + 8x - 21y - 28 (6xy + 8x) + (-21y - 28) Group2x(3y + 4) + (-7)(3y + 4) Factor GCF(3y +4) (2x - 7) Factor common binomial

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You must be able to recognize additive inverses!!!(3 - a and a - 3 are additive inverses because their sum is equal to zero.) Remember 3 - a = -1(a - 3).

Example 3:15x - 3xy + 4y - 20(15x - 3xy) + (4y - 20) Group3x(5 - y) + 4(y - 5) Factor GCF3x(-1)(-5 + y) + 4(y - 5) Notice additive inverses-3x(y - 5) + 4(y - 5) Simplify(y - 5) (-3x + 4) Factor common binomialRemember to check each problem by using FOIL.

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56 Factor 15ab - 3a + 10b - 2

A (5b - 1)(3a + 2)

B (5b + 1)(3a + 2)

C (5b - 1)(3a - 2)

D (5b + 1)(3a - 1)

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57 Factor 10m2n - 25mn + 6m - 15

A (2m-5)(5mn-3)

B (2m-5)(5mn+3)

C (2m+5)(5mn-3)

D (2m+5)(5mn+3)

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58 Factor 20ab - 35b - 63 +36a

A (4a - 7)(5b - 9)

B (4a - 7)(5b + 9)

C (4a + 7)(5b - 9)

D (4a + 7)(5b + 9)

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59 Factor a2 - ab + 7b - 7a

A (a - b)(a - 7)

B (a - b)(a + 7)

C (a + b)(a - 7)

D (a + b)(a + 7)

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Mixed Factoring


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Factor the Polynomial

Factor out GCF

2 Terms3 Terms

4 Terms

Differenceof Squares

Perfect SquareTrinomial

Factor theTrinomial

Group and Factorout GCF. Look for aCommon Binomial

Check each factor to see if it can be factored again.If a polynomial cannot be factored, then it is called prime.

Summary of Factoring

a = 1 a = 1

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A

B

C

D

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61 Factor completely

A

B

C

D prime polynomial

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62 Factor

A

B

C

D prime polynomial

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64 Factor

A

B

C

D Prime Polynomial

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Solving Equations by Factoring


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Given the following equation, what conclusion(s) can be drawn?ab = 0

Since the product is 0, one of the factors, a or b, must be 0.This is known as the Zero Product Property.

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Given the following equation, what conclusion(s) can be drawn?(x - 4)(x + 3) = 0

Since the product is 0, one of the factors must be 0. Therefore, either x - 4 = 0 or x + 3 = 0. x - 4 = 0 or x + 3 = 0 + 4 + 4 - 3 - 3 x = 4 or x = -3

Therefore, our solution set is {-3, 4}. To verify the results, substitute each solution back into the original equation.(x - 4)(x + 3) = 0

(-3 - 4)(-3 + 3) = 0(-7)(0) = 0

0 = 0

To check x = -3: (x - 4)(x + 3) = 0(4 - 4)(4 + 3) = 0

(0)(7) = 00 = 0

To check x = 4:

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What if you were given the following equation? How would you solve it?We can use the Zero Product Property to solve it. How can we turn this polynomial into a multiplication problem? Factor it! Factoring yields: (x - 6)(x + 4) = 0By the Zero Product Property:

x - 6 = 0 or x + 4 = 0After solving each equation, we arrive at our solution:

{-4, 6}

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65 Choose all of the solutions to:

A

B

C

D

E

F

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A

B

C

D

E

F

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A

B

C

D

E

F

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68 A ball is thrown with its height at any time given by When does the ball hit the ground?

A -1 seconds

B 0 seconds

C 9 seconds

D 10 seconds

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Documents

Factors Greatest Common Factorscontent.njctl.org/courses/math/archived-coursesunits-prior-to-comm… · The first step in factoring is to determine its greatest monomial factor. If