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Algebra I

Polynomials

www.njctl.org 2013-07-31

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Table of Contents

· Definitions of Monomials, Polynomials and Degrees · Adding and Subtracting Polynomials · Multiplying Polynomials · Mulitplying a Polynomial by a Monomial · Special Binomial Products · Solving Equations · Factors and GCF · Factoring out GCF's · Factoring 4 Term Polynomials · Identifying & Factoring x

2+ bx + c

· Factoring Using Special Patterns · Factoring Trinomials ax

2 + bx + c

· Mixed Factoring · Solving Equations by Factoring

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Definitions of Monomials, Polynomials and Degrees

Return to Table of Contents

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A monomial is a one- term expression formed by a number, a variable, or the product of numbers and variables. Examples of monomials....

81y4z

17x

2

4x

2 8

m n 3

rt 6

32,457

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a + b - 5 5 x + 7

x2(5 + 7y)

6+ 5rs 7x

3

y

5

• 4

Monomials

Drag the following terms into the correct sorting box. If you sort correctly, the term will be visible. If you sort incorrectly, the term will disappear.

48x2yz 3 4 ( 5 a

2

b c

2

) t 1 6

• 12

15 xy4 7

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A polynomial is an expression that contains two or more monomials. Examples of polynomials... .

5+ a

2

8 x

3

+ x

2

c 2+ d

8a3- 2b

2

4c- mn3

rt 6 a4b 15 +

7+ b+ c

2+ 4d 3

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Degrees of Monomials

The degree of a monomial is the sum of the exponents of its variables. The degree of a nonzero constant such as 5

• r 12 is 0. The constant 0 has no degree.

Examples:

1) The degree of 3x is? 1 The variable x has a degree 1. 2) The degree of -6x3y is? 4 The x has a power of 3 and the y has a power of 1, so the degree is 3+1 =4. 3) The degree of 9 is? 0 A constant has a degree 0, because there is no variable.

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1 What is the degree of ?

A B

1

C

2

D

3

Pull Pull

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2 What is the degree of ?

A B

1

C

2

D

3

Pull Pull

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3 What is the degree of 3 ?

A B

1

C

2

D

3

Pull Pull

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4 What is the degree of ?

Pull Pull

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Degrees of Polynomials

Example: Find degree of the polynomial 4x 3y2 - 6xy2 + xy. The monomial 4x3y2 has a degree of 5, the monomial 6xy2 has a degree of 3, and the monomial xy has a degree of 2. The highest degree is 5, so the degree of the polynomial is 5. The degree of a polynomial is the same as that of the term with the greatest degree.

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Find the degree of each polynomial 1) 3 2) 12c3 3) ab 4) 8s4t 5) 2 - 7n 6) h4 - 8t 7) s3 + 2v2y2 - 1 Answers: 1) 0 2) 3 3) 2 4) 5 5) 1 6) 4 7) 4

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5 What is the degree of the following polynomial:

A

3

B

4

C

5

D

6

Pull Pull

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6 What is the degree of the following polynomial:

A

3

B

4

C

5

D

6

Pull Pull

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7 What is the degree of the following polynomial:

A

3

B

4

C

5

D

6

Pull Pull

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8 What is the degree of the following polynomial:

A

3

B

4

C

5

D

6

Pull Pull

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Adding and Subtracting Polynomials

Return to Table of Contents

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Standard Form

The standard form of an equation is to put the terms in order from highest degree to the lowest degree. Standard form is commonly excepted way to write polynomials. Example: is in standard form. Put the following equation into standard form:

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Monomials with the same variables and the same power are like terms. Like Terms Unlike Terms 4x and -12x

• 3b and 3a

x3y and 4x3y 6a2b and -2ab

2

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Combine these like terms using the indicated operation.

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9 Simplify

A B C D Pull Pull

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10 Simplify

A B C D Pull Pull

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11 Simplify

A B C D Pull Pull

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To add polynomials, combine the like terms from each polynomial. To add vertically, first line up the like terms and then add. Examples: (3x2 +5x -12) + (5x 2 -7x +3) (3x

4 -5x) + (7x 4 +5x2 -14x)

line up the like terms line up the like terms 3x2 + 5x - 12 3x

4 -5x

(+)5x2 - 7x + 3 (+) 7x4 +5x2 - 14x

8x2 - 2x - 9 10x 4 +5x2 - 19x

=

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We can also add polynomials horizontally. (3x2 + 12x - 5) + (5x

2 - 7x - 9)

Use the communitive and associative properties to group like terms. (3x

2 + 5x2) + (12x + -7x) + (-5 + -9)

8x

2 + 5x - 14

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12 Add

A B C D Pull Pull

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13 Add

A B C D Pull Pull

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14 Add

A B C D Pull Pull

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15 Add

A B C D Pull Pull

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16 Add

A B C D Pull Pull

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To subtract polynomials, subtract the coefficients of like terms. Example:

• 3x - 4x = -7x

13y - (-9y) = 22y 6xy - 13xy = -7xy

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We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to adding -1. Distribute the -1 and then follow the rules for adding polynomials

(3x2 +4x -5) - (5x 2 -6x +3) (3x2+4x-5) +(-1) (5x2-6x+3) (3x2+4x-5) + (-5x 2+6x-3) 3x 2 + 4x - 5 (+) -5x2 - 6x + 3

• 2x

2 +10x - 8

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We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to adding -1. Distribute the -1 and then follow the rules for adding polynomials

(4x3 -3x -5) - (2x3 +4x2 -7) (4x3 -3x -5) +(-1)(2x3 +4x2 -7) (4x3 -3x -5) + (-2x 3 -4x2 +7) 4x 3 - 3x - 5

(+) -2x3 - 4x2 + 7

2x 3 - 4x2 - 3x + 2

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We can also subtract polynomials horizontally. (3x2 + 12x - 5) - (5x 2 - 7x - 9) Change the subtraction to adding a negative one and distribute the negative one. (3x2 + 12x - 5) +(-1)(5x 2 - 7x - 9) (3x2 + 12x - 5) + (-5x 2 + 7x + 9) Use the communitive and associative properties to group like terms. (3x2 +-5x2) + (12x +7x) + (-5 +9)

• 2x2 + 19x + 4

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17 Subtract

A B C D Pull Pull

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18 Subtract

A B C D Pull Pull

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19 Subtract

A B C D Pull Pull

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20 Subtract

A B C D Pull Pull

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21 Subtract

A B C D Pull Pull

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22 What is the perimeter of the following figure? (answers are in units)

A B C D Pull Pull

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Multiplying a Polynomial by a Monomial

Return to Table of Contents

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Find the total area of the rectangles.

3 5 8 4 square units square units

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To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication.

Examples: Simplify.

• 2x

(5x2 - 6x + 8)

• 2x

(5x2 + - 6x + 8) (- 2x )(5x2) + (

• 2x

)(- 6x) + (

• 2x

)(8)

• 10x

3 + 12x 2 + - 16x

• 10x

3 + 12x 2 - 16x

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To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication.

Examples: Simplify.

• 3x

2(- 2x 2 + 3x - 12)

• 3x

2(- 2x 2 + 3x + - 12)

(- 3x

2)(- 2x 2) + (

• 3x

2)(3x) + (

• 3x

2)(- 12)

6x4 + - 9x

3 + 36x 2

6x4 - 9x

3 + 36x 2

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12x4y3 - 15x

3y4 + 24x 2y5

Let's Try It! Multiply to simplify. 1.

• 2. 4x2(5x2 - 6x - 3)
• 3. 3xy(4x

3y2 - 5x 2y3 + 8xy 4)

• 2x 4 + 4x 3 - 7x2

20x4 - 24x

3 - 12x

Slide to check. Slide to check.

Slide to check.

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23What is the area of the rectangle shown?

A B C D x2 x2 + 2x + 4 Pull Pull

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24

A B C D

6x 2 + 8x - 12 6x 2 + 8x 2 - 12 6x 2 + 8x 2 - 12x 6x 3 + 8x 2 - 12x Pull Pull

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25

A B C D Pull Pull

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26

A B C D Pull Pull

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27Find the area of a triangle (A=1/2bh) with a base of 4x and a height of 2x - 8. All answers are in square units.

A B C D Pull Pull

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Multiplying Polynomials

Return to Table of Contents

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Find the total area of the rectangles.

5 8 2 6

sq.units

Area of the big rectangle Area of the horizontal rectangles Area of each box

(2 + 6) (5 + 8 ) = 2 (5 + 8) + 6 (5 + 8) = 2(5) + 2(8) + 6(5) + 6(8) = 10 + 16 + 30 + 48 = 148

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sq.units

Let us observe the work from the previous example, From to , we changed the problem so that instead of a polynomial times a polynomial, we now have a monomial times a polynomial. Use this to help solve the next example. (2 + 6) (5 + 8 ) = 2 (5 + 8) + 6 (5 + 8) = 2(5) + 2(8) + 6(5) + 6(8) = 10 + 16 + 30 + 48 = 148

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Find the total area of the rectangles.

2x 4 x 3

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To multiply a polynomial by a polynomial, you multiply each term of the first polynomials by each term of the second. Then, add like terms. Example 1: Example 2:

(2x + 4y)(3x + 2y) 2x(3x + 2y) + 4y(3x + 2y) 2x(3x) + 2x(2y) + 4y(3x) + 4y(2y) 6x 2 + 4xy + 12xy + 8y

2

6x 2 + 16xy + 8y

2

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The FOIL Method can be used to remember how multiply two binomials. To multiply two binomials, find the sum of .... First terms Outer terms Inner Terms Last Terms Example: First Outer Inner Last

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x2 - 7x + 12 2x3 + x

2 - 14x - 16

Try it! Find each product. 1) (x - 4)(x - 3) 2) (x + 2)(2x

2 - 3x - 8)

Slide to check. Slide to check.

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8x2 - 2xy - 15y

2

x4 + x

3 - 8x 2 + 24x - 24

3) (2x - 3y)(4x + 5y) 4) (x2 + 3x - 6)(x

2 - 2x + 4)

Try it! Find each product.

Slide to check. Slide to check.

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28What is the total area of the rectangles shown?

A B C D

4x 5 2x 4

Pull Pull

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29

A B C D Pull Pull

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30

A B C D Pull Pull

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31

A B C D Pull Pull

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32

A B C D Pull Pull

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33Find the area of a square with a side of

A B C D Pull Pull

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34What is the area of the rectangle (in square units)?

A B C D

3x2 + 5x + 2 3x2 + 6x + 2 3x2 - 6x + 2 3x2 - 5x +2

Pull Pull

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How would we find the area of the shaded region? Shaded Area = Total area - Unshaded Area

• sq. units

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35What is the area of the shaded region (in sq. units)?

A B C D 11x 2 + 3x - 8 7x 2 + 3x - 9 7x 2 - 3x - 10 11x 2 - 3x - 8 Pull Pull

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36What is the area of the shaded region (in square units)?

A B C D 2x 2 - 2x - 8 2x 2 - 4x - 6 2x 2 - 10x - 8 2x 2 - 6x - 4 Pull Pull

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Special Binomial Products

Return to Table of Contents

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Square of a Sum

(a + b)

2

(a + b)(a + b) a2 + 2ab + b

2

The square of a + b is the square of a plus twice the product of a and b plus the square of b.

Example: (5x + 3)

2

(5x + 3)(5x + 3) 25x

2 + 30x + 9

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Square of a Difference

(a - b)

2

(a - b)(a - b) a2 - 2ab + b

2

The square of a - b is the square of a minus twice the product of a and b plus the square of b. Example:

(7x - 4)

2

(7x - 4)(7x - 4) 49x

2 - 56x + 16

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Product of a Sum and a Difference

(a + b)(a - b) a2 + - ab + ab + - b

2

Notice the

• ab

and ab a2 - b

2 equals 0.

The product of a + b and a - b is the square of a minus the square of b.

Example: (3y - 8)(3y + 8) Remember the inner and 9y2 - 64

• uter terms equals 0.

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Try It! Find each product.

• 1. (3p + 9)

2

9p2 + 54p + 81

• 2. (6 - p)

2

36 - 12p + p

2

• 3. (2x - 3)(2x + 3)

4x2 - 9 Slide to check.

Slide to check. Slide to check.

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37

A B C D x2 + 25 x2 + 10x + 25 x2 - 10x + 25 x2 - 25 Pull Pull

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38

A B C D Pull Pull

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39What is the area of a square with sides 2x + 4?

A B C D Pull Pull

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40

A B C D Pull Pull

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Solving Equations

Return to Table of Contents

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Given the following equation, what conclusion(s) can b 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 . Slide 81 / 211

Zero Product Property

Rule: If ab=0, then either a=0 or b=0

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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

• r x + 3 = 0

. x - 4 = 0

• r

x + 3 = 0 + 4 + 4

• 3 - 3

x = 4

• r

x = - 3 Therefore, our solution set is {- 3, 4}. To verify the results, substitut 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?

(x - 6)(x + 4) = 0 By the Zero Product Property: x - 6 = 0 or x + 4 = 0 x = 6 x = - 4 After solving each equation, we arrive at our solution: {- 4, 6}

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41Solve (a + 3)(a - 6) = 0.

A

{3 , 6}

B

{-3 , -6}

C

{-3 , 6}

D

{3 , -6}

Pull Pull

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42Solve (a - 2)(a - 4) = 0.

A

{2 , 4}

B

{-2 , -4}

C

{-2 , 4}

D

{2 , -4}

Pull Pull

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43Solve (2a - 8)(a + 1) = 0.

A

{-1 , -16}

B

{-1 , 16}

C

{-1 , 4}

D

{-1 , -4}

Pull Pull

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Factors and Greatest Common Factors

Return to Table of Contents

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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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44 What is the GCF of 12 and 15?

Pull Pull

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45 What is the GCF of 24 and 48?

Pull Pull

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46 What is the GCF of 72 and 54?

Pull Pull

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47 What is the GCF of 70 and 99?

Pull Pull

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

Pull Pull

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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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49 What is the GCF of

A B C D

and ?

Pull Pull

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50 What is the GCF of

A B C D

and ?

Pull Pull

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51 What is the GCF of

A B C D

and

?

and

Pull Pull

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52 What is the GCF of

A B C D

and

?

and

Pull Pull

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Factoring

• ut GCFs

Return to Table of Contents

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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 - 15x

3 + 3x 2

3x 2 (2x 2 - 5x + 1) GCF: 3x2 Reduce each term of the polynomial dividing by the GCF Find the GCF 3x 2 3x 2 3x 2 3x 2 3x 2 6x 4 15x 3

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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 - 7m

2n2

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) +

( ( Slide 104 / 211

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

• pposites of each other.

For example: (x - y) = - (y - x) because

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

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

True False Pull Pull

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

True False Pull Pull

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

True False Pull Pull

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

True False Pull Pull

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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 Example 3 Factor each polynomial. a)

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

n(n - 3) - 7(3 - n) (n - 3)(n + 7)

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 opposites of each other. For example: (x - y) = - (y - x) because Example 3 Factor each polynomial. b)

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

p(h - 1) + 4(1 - h) (h - 1)(p - 4)

GCF: Reduce each term of the polynomial dividing by the GCF Find the GCF

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57 If possible, Factor

A B C D

Already Simplified

Pull Pull

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58 If possible, Factor

A B C D

Already Simplified

Pull Pull

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59 If possible, Factor

A B C D

Already Simplified

Slide 115 / 211

60 If possible, Factor

A B C D

Already Simplified

Pull Pull

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61 If possible, Factor

A B C D

Already Simplified

Pull Pull

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Identifying & Factoring: x2 + bx + c

Return to Table of Contents

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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 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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62 Choose all of the descriptions that apply to:

A

Quadratic

B

Linear

C

Constant

D

Trinomial

E

Binomial

F

Monomial

Pull Pull

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63 Choose all of the descriptions that apply to:

A

Quadratic

B

Linear

C

Constant

D

Trinomial

E

Binomial

F

Monomial

Pull Pull

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64 Choose all of the descriptions that apply to:

A

Quadratic

B

Linear

C

Constant

D

Trinomial

E

Binomial

F

Monomial

Pull Pull

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65 Choose all of the descriptions that apply to:

A

Quadratic

B

Linear

C

Constant

D

Trinomial

E

Binomial

F

Monomial

Pull Pull

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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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To Factor a Trinomial with a Lead Coefficient of 1

Recognize the pattern: Factors of 6 have the same signs. Factors of 6 add to +5. Both factors must be positive.

Factors of 6 Sum to 5? 1, 6 7 2, 3 5

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To Factor a Trinomial with a Lead Coefficient of 1

Recognize the pattern: Factors of 6 have the same signs. Factors of 6 add to

• 7. Both factors

must be negative.

Factors of 6 Sum to -7?

• 1, -6
• 7
• 2, -3
• 5

Slide 127 / 211

(x - 8)(x - 1)

Examples:

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66 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

Pull Pull

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67 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

Pull Pull

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68 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)

Pull Pull

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69 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)

Pull Pull

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70 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)

Pull Pull

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71 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 - 2)

F

(x - 4)(x - 3)

Pull Pull

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To Factor a Trinomial with a Lead Coefficient of 1

Recognize the pattern: Factors of 6 have the

• pposite signs.

Factors of 6 add to

• 5. Larger factor

must be negative.

Factors of 6 Sum to -5? 1, -6

• 5

2, -3

• 1

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To Factor a Trinomial with a Lead Coefficient of 1

Recognize the pattern: Factors of 6 have the opposite signs. Factors of 6 add to +1. Larger factor must be positive.

Factors of 6 Sum to 1?

• 1, 6

5

• 2, 3

1

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Examples

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72 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

Pull Pull

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73 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

Pull Pull

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74 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)

Pull Pull

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75 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

unable to factor using this method

Pull Pull

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76 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)

Pull Pull

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Mixed Practice

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77 Factor the following

A

(x - 2)(x - 4)

B

(x + 2)(x + 4)

C

(x - 2)(x +4)

D

(x + 2)(x - 4)

Pull Pull

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78 Factor the following

A

(x - 3)(x - 5)

B

(x + 3)(x + 5)

C

(x - 3)(x +5)

D

(x + 3)(x - 5)

Pull Pull

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79 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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80 Factor the following

A

(x - 2)(x - 5)

B

(x + 2)(x + 5)

C

(x - 2)(x +5)

D

(x + 2)(x - 5)

Pull Pull

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Steps for Factoring a Trinomial

1) See if a monomial can be factored out. 2) Need 2 numbers that multiply to the constant 3) and add to the middle number. 4) Write out the factors.

There is no common monomial,so factor:

STEP 1 STEP 2 STEP 3 STEP 4

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There is no common monomial,so factor:

STEP 1 STEP 2 STEP 3 STEP 4 Steps for Factoring a Trinomial

1) See if a monomial can be factored out. 2) Need 2 numbers that multiply to the constant 3) and add to the middle number. 4) Write out the factors.

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Factor out

Factor: STEP 2 STEP 3 STEP 4 STEP 1 Steps for Factoring a Trinomial

1) See if a monomial can be factored out. 2) Need 2 numbers that multiply to the constant 3) and add to the middle number. 4) Write out the factors.

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Factor out

Factor: STEP 4 STEP 3 STEP 2 STEP 1 Steps for Factoring a Trinomial

1) See if a monomial can be factored out. 2) Need 2 numbers that multiply to the constant 3) and add to the middle number. 4) Write out the factors.

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81 Factor completely:

A B C D Pull Pull

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82 Factor completely:

A B C D Pull Pull

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83 Factor completely:

A B C D Pull Pull

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84 Factor completely:

A B C D Pull Pull

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85 Factor completely:

A B C D Pull Pull

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Factoring Using Special Patterns

Return to Table of Contents

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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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Factoring Perfect Square Trinomials.

Once a Perfect Square Trinomial has been identified, it factors following the form: (sq rt of the first term sign of the middle term sq rt of the third term)2 Examples:

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86 Factor

A B C D Not a perfect Square Trinomial Pull Pull

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87 Factor

A B C D Not a perfect Square Trinomial Pull Pull

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88 Factor

A B C D Not a perfect Square Trinomial Pull Pull

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Difference of Squares

The Product of a Sum and a Difference is a difference of Squares. A Difference of Squares is recognizable by seeing each term in the binomial are perfect squares and the operation is subtraction.

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Examples of Difference of Squares

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Is the binomial a difference of squares?

Drag the Difference of Squares binomials into the Box. Only Difference of Squares will remain visible.

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Factoring a Difference of Squares

Once a binomial is determined to be a Difference of Squares, it factors following the pattern: (sq rt of 1st term - sq rt of 2nd term)(sq rt of 1st term + sq rt of 2nd term) Examples:

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89 Factor

A B C D

Not a Difference of Squares

Pull Pull

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90 Factor

A B C D

Not a Difference of Squares

Pull Pull

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91 Factor

A B C D

Not a Difference of Squares

Pull Pull

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92 Factor using Difference of Squares:

A B C D

Not a Difference of Squares

Pull Pull

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93 Factor

A B C D Pull Pull

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Factoring Trinomials: ax2 + bx + c

Return to Table of Contents

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How to factor a trinomial of the form ax² + bx + c.

Example: Factor 2d² + 15d + 18 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.

2b - b- 10

a = 2 , c = - 10, and b = - 1 Since a times c is negative, and b is negative we need to find two factors with opposite signs whose product is - 20 an that add up to - 1. Since the sum is negative, larger factor of - 2 must be negative. Factors of - 20 Sum = - 1? 1, - 20 2, - 10 4, - 5 1 + - 20 = - 19 2 + - 10 = - 8 4 + - 5 = - 1

2

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Factor 6y² - 13y - 5

Pull Pull

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

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94 Factor

A B C D

Prime Polynomial

Pull Pull

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95 Factor

A B C D

Prime Polynomial

Pull Pull

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96 Factor

A B C D

Prime Polynomial

Pull Pull

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Factoring 4 Term Polynomials

Return to Table of Contents

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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 an 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 z 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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97 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)

Pull Pull

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98 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)

Pull Pull

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99 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)

Pull Pull

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100 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)

Pull Pull

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Mixed Factoring

Return to Table of Contents

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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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3r 3 - 9r 2 + 6r 3r(r 2 - 3r + 2) 3r(r - 1)(r - 2)

Examples

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101 Factor completely:

A B C D Pull Pull

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102 Factor completely

A B C D

prime polynomial

Pull Pull

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103 Factor

A B C D prime polynomial Pull Pull

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104 Factor completely

A B C D 10w2(x2 -10x +100) Pull Pull

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105 Factor

A B C D Prime Polynomial Pull Pull

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Solving Equations by Factoring

Return to Table of Contents

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Given the following equation, what conclusion(s) can b 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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Recall ~ Given the following equation, what conclusion(s) ca drawn?

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

• r x + 3 = 0

. x - 4 = 0

• r

x + 3 = 0 + 4 + 4

• 3 - 3

x = 4

• r

x = - 3 Therefore, our solution set is {- 3, 4}. To verify the results, substitut 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? Fac 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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Solve Recall the Steps for Factoring a Trinomial

1) See if a monomial can be factored out. 2) Need 2 numbers that multiply to the constant 3) and add to the middle number. 4) Write out the factors.

Now...

1) Set each binomial equal to zero. 2) Solve each binomial for the variable.

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Zero Product rule works only when the product of factors equals zero. If the equation equals some value other than zero, subtract to make one side of the equation zero.

Example

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106 Choose all of the solutions to:

A B C D E F Pull Pull

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107 Choose all of the solutions to:

A B C D E F Pull Pull

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108 Choose all of the solutions to:

A B C D E F Pull Pull

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Application~ A science class launches a toy rocket. The teacher tells the class that the height of the rocket at any given time is h = -16t2 + 320t. When will the rocket hit the ground? When the rocket hits the ground, its height is 0. So h=0 which can be substituted into the equation: The rocket had to hit the ground some time after launching. The rocket hits the ground in 20 seconds. The 0 is an extraneous (extra) answer.

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109 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

Pull Pull

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