algebra 7-6 notes: choosing a factoring method (pp 498 …7_6... · algebra 7-6 notes: choosing a...

Algebra 7-6 Notes: Choosing a factoring Method (pp 498-503) Page ! of !1 8

Attendance Problems. Factor each trinomial. 1. x2 + 13x + 40 2. 5x2 – 18x – 8 !!!!!!!!!!!!

3. Factor the perfect-square trinomial 16x2 + 40x + 25 !!4. Factor 9x2 – 25y2 using the difference of two squares. !• I can choose an appropriate method for factoring a polynomial. • I can combine methods for factoring a polynomial. !!Common Core Standard: CC.9-12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4–y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2). !Video Example 1: Determine whether the each expression is completely factored.

A) ! B) !

!!!!!

2w(w2 + 5w + 7) 6x +10( ) x2 + 9( )


!

! Example 1. Tell whether each expression is completely factored. If not, factor it.

A. 3x2(6x – 4) B. (x2 + 1)(x – 5) !!!!!!

Choosing a Factoring Method

ObjectivesChoose an appropriate method for factoring a polynomial.

Combine methods for factoring a polynomial.

Why learn this?You will need to factor polynomials to solve quadratic equations, which have many applications in physics. (See Exercise 42.)

The height of a leaping ballet dancer can be modeled by a quadratic polynomial. Solving an equation that contains that polynomial may require factoring the polynomial.

Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further.

1E X A M P L E Determining Whether an Expression Is Completely Factored

Tell whether each expression is completely factored. If not, factor it.

A 2x ( x 2 + 4)

2x ( x 2 + 4) Neither 2x nor x 2 + 4 can be factored further.

2x + 6 can be further factored.

Factor out 2, the GCF of 2x and 6.

2x ( x 2 + 4) is completely factored.

B (2x + 6) (x + 5)

(2x + 6) (x + 5)

2 (x + 3) (x + 5)

2 (x + 3) (x + 5) is completely factored.

Tell whether each expression is completely factored. If not, factor it.

1a. 5 x 2 (x - 1) 1b. (4x + 4) (x + 1)

To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below to factor a polynomial completely.

Factoring Polynomials

Step 1 Check for a greatest common factor.

Step 2 Check for a pattern that fits the difference of two squares or a perfect-square trinomial.

Step 3 To factor x 2 + bx + c, look for two numbers whose sum is b and whose product is c.

To factor a x 2 + bx + c, check factors of a and factors of c in the binomial factors. The sum of the products of the outer and inner terms should be b.

Step 4 Check for common factors.

x 2 + 4 is a sum of squares, and cannot be factored.

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7-6CC.9-12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it.

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T eac h i ng T ra ns pa r e ncy

Copyright © by Holt McDougal. Holt McDougal Algebra 1All rights reserved.

Choosing a Factoring Method

Factoring Polynomials

Step 1 Check for a greatest common factor.

Step 2 Check for a pattern that fits the difference of two squares or a perfect-square trinomial.

Step 3 To factor x 2 + bx + c, look for two numbers whose sum is b and whose product is c.

To factor a x 2 + bx + c, check factors of a and factors of c in the binomial factors. The sum of the products of the outer and inner terms should be b.

Step 4 Check for common factors.

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Guided Practice. Tell whether each expression is completely factored. If not, factor it.

5. 5x2(x – 1) 6. (4x + 4)(x + 1) !!!!!Video Example 2: Factor ! completely. !!!!!!

! Example 2. Factor each polynomial completely.

A. 10x2 + 48x + 32 B. 8x6y2 – 18x2y2 !!!!!!!!!!!!!

−3ab2 + 30ab − 75a

2E X A M P L E Factoring by GCF and Recognizing Patterns

Factor -2x y 2 + 16xy - 32x completely. Check your answer.

-2x y 2

+ 16xy - 32x-2x ( y 2 - 8y + 16) Factor out the GCF. y 2 - 8y + 16 is a perfect-

square trinomial of the form a 2 - 2ab + b 2 .a = y, b = 4-2x (y - 4) 2

Check -2x (y - 4) 2 = -2x ( y 2 - 8y + 16) = -2x y 2 + 16xy - 32x ✓

Factor each polynomial completely. Check your answer. 2a. 4 x 3 + 16 x 2 + 16x 2b. 2 x 2 y - 2 y 3

If none of the factoring methods work, the polynomial is unfactorable.

3E X A M P L E Factoring by Multiple Methods

Factor each polynomial completely.

A 2 x 2 + 5x + 4 2 x 2 + 5x + 4 The GCF is 1 and there is no pattern.

a = 2 and c = 4; Outer + Inner = 5 ( x + ) ( x + )

Factors of 2 Factors of 4 Outer + Inner

1 and 2

1 and 2

1 and 2

1 and 4

4 and 1

2 and 2

1 (4) + 2 (1) = 6

1 (1) + 2 (4) = 9

1 (2) + 2 (2) = 6

✗

✗

✗

2 x 2 + 5x + 4 is unfactorable.

B 3 n 4 - 15 n 3 + 12 n 2 3 n 2 ( n 2 - 5n + 4) Factor out the GCF. There is no pattern.

b = -5 and c = 4; look for factors of 4 whose sum is -5.

The factors needed are -1 and -4.

Factor out the GCF. There is no pattern.a = 2 and c = 10; Outer + Inner = 9

(n + ) (n + )

Factors of 4 Sum-1 and -4 -5 ✓

3 n 2 (n - 1) (n - 4)

C 4 x 3 + 18 x 2 + 20x 2x (2 x 2 + 9x + 10)

( x + ) ( x + )


1 and 2

1 and 2

1 and 2

1 and 10

10 and 1

2 and 5

1 (10) + 2 (1) = 12

1 (11) + 2 (10) = 21

1 (5) + 2 (2) = 9

✗

✗

✓

(x + 2) (2x + 5)

2x (x + 2) (2x + 5)

For a polynomial ofthe form a x 2 + bx + c,if there are no integers whose sum is b and whose product is ac, then the polynomial is unfactorable.

7-6 Choosing a Factoring Method 499

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Guided Practice: Factor each polynomial completely. 7) ! 8) ! !!!!!!!!

Video Example 3: Factor each polynomial completely. A) ! B) ! !!!!!!!!!!!!

C) ! D)! !!!!!!!!!

4x3 +16x2 +16x 2x2y − 2y3

5w2 + 3w + 9 5y4 − 25y3 + 30y2

15x3 + 66x2 + 24x z − z5


!

! !

2E X A M P L E Factoring by GCF and Recognizing Patterns

Factor -2x y 2 + 16xy - 32x completely. Check your answer.

-2x y 2

+ 16xy - 32x-2x ( y 2 - 8y + 16) Factor out the GCF. y 2 - 8y + 16 is a perfect-

square trinomial of the form a 2 - 2ab + b 2 .a = y, b = 4-2x (y - 4) 2

Check -2x (y - 4) 2 = -2x ( y 2 - 8y + 16) = -2x y 2 + 16xy - 32x ✓

Factor each polynomial completely. Check your answer. 2a. 4 x 3 + 16 x 2 + 16x 2b. 2 x 2 y - 2 y 3

If none of the factoring methods work, the polynomial is unfactorable.

3E X A M P L E Factoring by Multiple Methods

Factor each polynomial completely.

A 2 x 2 + 5x + 4 2 x 2 + 5x + 4 The GCF is 1 and there is no pattern.

a = 2 and c = 4; Outer + Inner = 5 ( x + ) ( x + )


1 and 2

1 and 2

1 and 2

1 and 4

4 and 1

2 and 2

1 (4) + 2 (1) = 6

1 (1) + 2 (4) = 9

1 (2) + 2 (2) = 6

✗

✗

✗

2 x 2 + 5x + 4 is unfactorable.

B 3 n 4 - 15 n 3 + 12 n 2 3 n 2 ( n 2 - 5n + 4) Factor out the GCF. There is no pattern.

b = -5 and c = 4; look for factors of 4 whose sum is -5.

The factors needed are -1 and -4.

Factor out the GCF. There is no pattern.a = 2 and c = 10; Outer + Inner = 9

(n + ) (n + )

Factors of 4 Sum-1 and -4 -5 ✓

3 n 2 (n - 1) (n - 4)

C 4 x 3 + 18 x 2 + 20x 2x (2 x 2 + 9x + 10)

( x + ) ( x + )


1 and 2

1 and 2

1 and 2

1 and 10

10 and 1

2 and 5

1 (10) + 2 (1) = 12

1 (11) + 2 (10) = 21

1 (5) + 2 (2) = 9

✗

✗

✓

(x + 2) (2x + 5)

2x (x + 2) (2x + 5)

For a polynomial ofthe form a x 2 + bx + c,if there are no integers whose sum is b and whose product is ac, then the polynomial is unfactorable.

7-6 Choosing a Factoring Method 499

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D p 5 - p

p ( p 4 - 1) Factor out the GCF.

p4 - 1 is a difference of two squares.


p ( p 2 + 1) ( p 2 - 1)

p ( p 2 + 1) (p + 1) (p - 1)

Factor each polynomial completely. Check your answer. 3a. 3 x 2 + 7x + 4 3b. 2 p 5 + 10 p 4 - 12 p 3 3c. 9 q 6 + 30 q 5 + 24 q 4 3d. 2 x 4 + 18

Any Polynomial—Look for the greatest common factor.

ab - ac = a (b - c) 6 x 2 y + 10x y 2 = 2xy (3x + 5y)

Binomials—Look for a difference of two squares.

a 2 - b 2 = (a + b) (a - b) x 2 - 9 y 2 = (x + 3y) (x - 3y)

Trinomials—Look for perfect-square trinomials and other factorable trinomials.

a 2 + 2ab + b 2 = (a + b) 2

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

x 2 + 4x + 4 = (x + 2) 2

x 2 - 2x + 1 = (x - 1) 2

x 2 + bx + c = (x + ) (x + )

a x 2 + bx + c = ( x + ) ( x + ) x 2 + 3x + 2 = (x + 1) (x + 2)

6 x 2 + 7x + 2 = (2x + 1) (3x + 2)

Polynomials of Four or More Terms—Factor by grouping.

ax + bx + ay + by = x (a + b) + y (a + b)

= (x + y) (a + b)

2 x 3 + 4 x 2 + x + 2 = (2 x 3 + 4 x 2 ) + (x + 2)

= 2 x 2 (x + 2) + 1 (x + 2)

= (x + 2) (2 x 2 + 1)

Methods to Factor Polynomials

THINK AND DISCUSS 1. Give an expression that includes a polynomial that is not completely

factored.

2. Give an example of an unfactorable binomial and an unfactorable trinomial.

3. GET ORGANIZED Copy the graphic organizer. Draw an arrow from each expression to the method you would use to factor it.




Example 3. Factor each polynomial completely. A. 9x2 + 3x – 2 B. 12b3 + 48b2 + 48b C. 4y2 + 12y – 72 !!!!!!!!!!!!

D. (x4 – x2) !!!!!!!!!!!!!


Guided Practice. Factor each polynomial completely. 9) ! 10) ! !!!!!!!!!!!!

11) ! 12) ! !!!!!!!!!!!!

3x2 + 7x + 4 2p5 +10p4 −12p3

9q6 + 30q5 + 24q4 2x4 +18


7-6 Choosing a Factoring Method: (p 501) 21-35 odd, 37, 42, 43-45, 48, 49, 51-53. !Question: Where do math teachers shop? !Answer: At the factor—y

D p 5 - p

p ( p 4 - 1) Factor out the GCF.



p ( p 2 + 1) ( p 2 - 1)

p ( p 2 + 1) (p + 1) (p - 1)

Factor each polynomial completely. Check your answer. 3a. 3 x 2 + 7x + 4 3b. 2 p 5 + 10 p 4 - 12 p 3 3c. 9 q 6 + 30 q 5 + 24 q 4 3d. 2 x 4 + 18

Any Polynomial—Look for the greatest common factor.

ab - ac = a (b - c) 6 x 2 y + 10x y 2 = 2xy (3x + 5y)

Binomials—Look for a difference of two squares.

a 2 - b 2 = (a + b) (a - b) x 2 - 9 y 2 = (x + 3y) (x - 3y)

Trinomials—Look for perfect-square trinomials and other factorable trinomials.

a 2 + 2ab + b 2 = (a + b) 2

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

x 2 + 4x + 4 = (x + 2) 2

x 2 - 2x + 1 = (x - 1) 2

x 2 + bx + c = (x + ) (x + ) a x 2 + bx + c = ( x + ) ( x + )

x 2 + 3x + 2 = (x + 1) (x + 2)

6 x 2 + 7x + 2 = (2x + 1) (3x + 2)

Polynomials of Four or More Terms—Factor by grouping.

ax + bx + ay + by = x (a + b) + y (a + b)

= (x + y) (a + b)

2 x 3 + 4 x 2 + x + 2 = (2 x 3 + 4 x 2 ) + (x + 2)

= 2 x 2 (x + 2) + 1 (x + 2)

= (x + 2) (2 x 2 + 1)

Methods to Factor Polynomials

THINK AND DISCUSS 1. Give an expression that includes a polynomial that is not completely

factored.

2. Give an example of an unfactorable binomial and an unfactorable trinomial.

3. GET ORGANIZED Copy the graphic organizer. Draw an arrow from each expression to the method you would use to factor it.



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