Factors 2 12 Factors Factors 3 13 and Unique Unique 14 4 - - PDF document

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Polynomials

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· 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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Factors and Greatest Common Factors

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number Bank Factors of 10 Factors of 15

Factors Unique to 15 Factors Unique to 10 Factors 10 and 15 have in common

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

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number Bank Factors of 12 Factors of 18

Factors Unique to 18 Factors Unique to 12 Factors 12 and 18 have 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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2 What is the GCF of 24 and 48?

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3 What is the GCF of 72 and 54?

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4 What is the GCF of 70 and 99?

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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 and and

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6 What is the GCF of

A B C D

and ?

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Factoring

• ut 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

• f 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 dividing by the GCF Find the GCF 3x2 3x2 3x2 3x2 3x2 6x4 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

• f this greatest monomial factor and a polynomial.

Example 1 Factor each polynomial. b)

4m3n - 7m2n2 m2n(4n - 7n) GCF: m2n Reduce each term of the polynomial dividing by 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 - 3 Reduce each term of the polynomial dividing by 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 dividing by the GCF Find the GCF

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In working with common binomial factors, look for factors that are

• pposites 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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15 If possible, Factor

A B C D

Already Simplified

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16 If possible, Factor

A B C D

Already Simplified

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17 If possible, Factor

A B C D

Already Simplified

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18 If possible, Factor

A B C D

Already Simplified

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19 If possible, Factor

A B C D

Already Simplified

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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.

Q u a d r a t i c t e r m . Linear term. Constant term.

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A quadratic polynomial in which b ≠ 0 and c ≠ 0 is called a quadratic 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

Quadratic

B

Linear

C

Constant

D

Trinomial

E

Binomial

F

Monomial

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22 Choose all of the descriptions that apply to:

A

Quadratic

B

Linear

C

Constant

D

Trinomial

E

Binomial

F

Monomial

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23 Choose all of the descriptions that apply to:

A

Quadratic

B

Linear

C

Constant

D

Trinomial

E

Binomial

F

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 + 4

x2 - 4x - 5 x2 + 4x - 12

Slide each polynomial from the circle to the correct expression.

x2 + 5x + 6 Answer Bank

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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

C

Bigger factor positive, the other negative

D

The bigger factor negative, the other positive

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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

C

Bigger factor positive, the other negative

D

The bigger factor negative, the other positive

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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

C

Bigger factor positive, the other negative

D

The bigger factor negative, the other positive

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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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37 Factor the following

A

(x - 3)(x - 5)

B

(x + 3)(x + 5)

C

(x - 3)(x +5)

D

(x + 3)(x - 5)

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38 Factor the following

A

(x - 3)(x - 4)

B

(x + 3)(x + 4)

C

(x +2)(x +6)

D

(x + 1)(x+12)

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39 Factor the following

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: STEP 4 STEP 3 STEP 2 STEP 1

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40 Factor completely:

A B C D

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41 Factor completely:

A B C D

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42 Factor completely:

A B C D

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43 Factor completely:

A B C D

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44 Factor completely:

A B C D

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Factoring Using Special Patterns

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When we were multiplying polynomials we had special patterns.

Square of Sums Difference of Sums Product 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 Trinomials

The 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 D Not a perfect Square Trinomial

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47 Factor

A B C D Not a perfect Square Trinomial

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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 D

Not a Difference of Squares

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50 Factor

A B C D

Not a Difference of Squares

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51 Factor using Difference of Squares:

A B C D

Not a Difference of Squares

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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, 36 2, 18 3, 12 1 + 36 = 37 2 + 18 = 20 3 + 12 = 15 Factors of 36 Sum = 15?

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Factor. 15x² - 13x + 2

a = 15 and c = 2, but b = -13 Since both a and c are positive, and b is negative we need to 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 Term Polynomials

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Polynomials with four terms like ab - 4b + 6a - 24, can be factored 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) Group 2x(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) Group 3x(5 - y) + 4(y - 5) Factor GCF 3x(-1)(-5 + y) + 4(y - 5) Notice additive inverses

• 3x(y - 5) + 4(y - 5)

Simplify (y - 5) (-3x + 4) Factor common binomial Remember 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 Terms 3 Terms

4 Terms

Difference

• f Squares

Perfect Square

Trinomial Factor the Trinomial Group and Factor

• ut GCF. Look for a

Common 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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60 Factor completely:

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) = 0 0 = 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) = 0 By the Zero Product Property: x - 6 = 0 or x + 4 = 0 After 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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66 Choose all of the solutions to:

A B C D E F

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67 Choose all of the solutions to:

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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