Algebra I Notes Unit 07b: Polynomials, Factoring and...

Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b

Alg I Unit 07b Notes PolynomialsFactorSpecial Products Page 1 of 18 3/18/2013

Note:

This unit can be used as needed (review or introductory) to practice factoring polynomials. This

will prepare students for solving quadratics and polynomial equations.

Math Background

Previously, you

Applied the laws of exponents and explored exponential functions

Identified and evaluated expressions involving exponents

Add, subtract, and multiply polynomials

In this unit you will study

Factor trinomials

Factor special products

Prepare to solve quadratics

You can use the skills in this unit to

Work with and solve practical applications such as area and free fall

Factor polynomials as products to develop methods of solving quadratics

Overall Big Ideas

There are many real-life applications where polynomial equations can be written to solve them.

Essential Questions

What are different ways to factor quadratic equations and which ways are most efficient?

Do all quadratics have real solutions?

Note: A file Algebra Unit 07 Practice – X Patterns can be useful to prepare students to quickly find sum and

product.



Teacher Note: The sample questions should be used to prepare for instruction.

Sample Questions

1. Which expression is equivalent to xc xb yc yb ?

A. x b y c

B. x c y b

C. x y b c

2. Which is equivalent to 2 44 9x y

A. 2

22 3x y

B. 2 22 3 2 3x y x y

C. 22 3 2 3 2 3x y x y x y

For questions 3-5, use the expression 4 4x y .

3. 2 2 2 2x y x y is equivalent to the given expression.

A. True

B. False

4. 2 2x y x y x y is equivalent to the given expression.

A. True

B. False

5. 3

x y x y is equivalent to the given expression.

A. True

B. False



6. Let 2 2 23x y and 6xy . What is the value of 2

x y ?

A. 9

B. 23

C. 29

D. 35

7. Which of these is NOT a factor of 212 6 90x x ?

A. 6

B. 2x

C. x + 3

D. 2x – 5

8. The expression 24 3x bx is factorable into two binomials. Which could NOT equal b?

A. –7

B. –1

C. 1

D. 11

9. Given 224 28 2x x c x q , where c and q are integers, what is the value of c?

A. 2

B. 7

C. 14

D. 49

10. If 7x is a factor of 22 11x x k , what is the value of k?

A. –21

B. –7

C. 7

D. 28



11. Factor 225 4x .

A. 5 2 5 2x x

B. 2

5 2x

C. The expression is not factorable with real coefficients.

12. Factor 29 16x .

A. 3 4 3 4x x

B. 2

3 4x

C. The expression is not factorable with real coefficients.

13. Which is a factor of 24 6 40x x ?

A. 2 5x

B. 2 5x

C. 2 4x

D. 2 4x

14. Which equation has roots of 4 and 6 ?

A. 4 6 0x x

B. 4 6 0x x

C. 4 6 0x x

D. 4 6 0x x

15. Which expression is equivalent to 2 3 40x x ?

A. 5 8x x

B. 5 8x x

C. 5 8x x

D. 5 8x x



16. Which expression is equivalent to 235 26 16x x ?

A. 7 2 5 8x x

B. 7 2 5 8x x

C. 7 8 5 2x x

D. 7 8 5 2x x

17. What value of c makes the expression 2 9y y c a perfect trinomial square?

A. –9

B. 9

2

C. 81

D. 81

4

18. What expression must the center cell of the table contain so that the sums of each row, each

column, and each diagonal are equivalent?

25 9x x 2 4x x 22 3 2x x 2 3 2x x 25 12x x 22 8x x 25 3 6x x 2 1x x

A. 22 5x x

B. 24 2 10x x

C. 26 3 15x x

19. Which is equivalent to 2 23 2x x y xy ?

A. 2 33 6x y xy

B. 3 23 2x y xy

C. 3 2 23 6x y x y

D. 4 39x y



20. Under what operations is the system of polynomials NOT closed?

A. addition

B. subtraction

C. multiplication

D. division

21. Which expression is equivalent to 2 26 4 3 5 8 7x x x x ?

A. 22 3 2x x

B. 22 11 2x x

C. 214 3 8x x

D. 214 11 8x x

22. Subtract:

2 29 5 6 3 4y y y y

A. 26 4 2y y

B. 26 4 10y y

C. 26 6 2y y

D. 26 6 10y y

23. Expand the expression 2

3 7x .

A. 29 42 49x x

B. 29 42 49x x

C. 29 49x

D. 29 49x



For questions 24-26, answer each with respect to the system of polynomials.

24. The system of polynomials is closed under subtraction.

A. True

B. False

25. The system of polynomials is closed under division.

A. True

B. False

26. The system of polynomials is closed under multiplication.

A. True

B. False

For questions 27-28, use the scenario below.

A rectangular playground is built such that its length is twice its width.

27. The area of the playground can be expressed as 2w2.

A. True

B. False

28. The perimeter of the playground can be expressed as 4w4.

A. True

B. False

w

= 2w



Notes

Skill: Factor polynomials connecting the arithmetic and algebraic processes.

A.SSE.3b Write expressions in equivalent forms to solve problems

3. Choose and produce an equivalent form of an expression to reveal and explain properties of the

quantity represented by the expression.★

b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the

function it defines.

Review: Greatest Common Factor (GCF)

Ex: Find the GCF of 45 and 60.

Recall: The GCF is the largest factor that the 2 numbers have in common.

We can write the prime factorization of each: 45 3 3 5 60 2 2 3 5

Now write all of the factors they have in common: 3 5 15

The GCF is 15.

Ex: Find the GCF of 26x y and

316x .

Write out each term as a product of factors: 26 2 3x y x x y

316 2 2 2 2x x x x

Write all of the factors they have in common: 22 2x x x

Factoring Using the Distributive Property

Ex: Factor the polynomial 22 8x x .

Step One: Find the GCF of the terms. 22 2 8 2 2 2x x x x x GCF = 2x

Step Two: Use the distributive property to factor the GCF out of the polynomial.

2 2 2

2 4

x x

x x

Ex: Factor the polynomial 3 2 2 214 21 7x y x y x y .

Step One: Find the GCF of the terms.

3

2 2

2

14 2 7

21 3 7

7 7

x y x x x y

x y x x y y

x y x x y

GCF = 27x y



Step Two: Use the distributive property to factor the GCF out of the polynomial.

2

2

7 2 3 1

7 2 3 1

x y x y

x y x y

Note: You can check your answers by multiplying using the distributive property.

Factoring by Grouping

Ex: Factor the polynomial 3 22 3 6x x x .

Step One: Group the first two terms and last two terms. 3 22 3 6x x x

Step Two: Factor the GCF from both sets of terms. 2 2 3 2x x x

Step Three: Factor the common factor using the distributive property. 22 3x x

Ex: Factor the polynomial 3 26 3 18n n n .

Step One: Group the first two terms and last two terms. 3 26 3 18n n n

Step Two: Factor the GCF from both sets of terms. 2 6 3 6n n n

Step Three: Factor the common factor using the distributive property. 26 3n n

You Try: Find the GCF and factor it out of the polynomial. 6 4 2 218 6 24x x y x y

QOD: How can you find the GCF of a variable expression without writing out all of the variables as factors?



Factoring a Quadratic Polynomial

Multiply x p x q using FOIL. 2 2x qx px pq x q p x pq

To factor 2x bx c into two binomials, x p x q , we must find values for p and q such that b q p ,

and c pq .

Ex: Factor 2 5 4x x .

Step One: Find values for p and q such that 4pq and 5p q .

We will list all of the factors of 4: 4 1 4 2 2 1 4 2 2

The two factors that have a sum of 5 are 1 and 4, so 1p and 4q .

Step Two: Write each factor x p x q 1 4x x

Note: Because multiplication is commutative, we could also write our answer as 4 1x x

Check your answer using FOIL!



We will list all of the factors of 24: 24 1 24 2 12 3 8 4 6

1 24 2 12 3 8 4 6

The two factors that have a sum of −10 are −4 and −6, so 4p and 6q .





We will list all of the factors of −9: 9 1 9 9 1 3 3

The two factors that have a sum of −8 are −9 and 1, so 9p and 1q .





Teacher Note: Check out the Notes for Unit 07 – Factor by Splitting the Middle Term on

www.rpdp.net

Factoring Quadratic Trinomials in the Form 2 , 1ax bx c a

Ex: Factor the trinomial 22 11 5x x .

Step One: Multiply ac. Find values for p and q such that pq ac , and p q b

10 1 10 2 5 1 10 2 5ac

The two factors that have a sum of 11 are 1 and 10.

Step Two: Split the middle term (bx) into two terms px qx . 22 1 10 5x x x

Step Three: Factor by grouping.

22 1 10 5

2 1 5 2 1

2 1 5

x x x

x x x

x x


Ex: Factor 26 11 2n n .


12 1 12 1 12 2 6 2 6 3 4 3 4ac

The two factors that have a sum of −11 are 1 and −12.

Step Two: Split the middle term (bx) into two terms px qx . 26 1 12 2n n n


26 1 12 2

6 1 2 6 1

6 1 2

n n n

n n n

n n


Ex: Factor 24 9 5t t .


20 1 20 1 20 2 10 2 10 4 5 4 5ac

The two factors that have a sum of −9 are −4 and −5.

Step Two: Split the middle term (bx) into two terms px qx . 24 4 5 5t t t

http://www.rpdp.net/




24 4 5 5

4 1 5 1

1 4 5

t t t

t t t

t t


Ex: Factor the trinomial 26 19 10x x .


60 1 60 2 30 3 20 4 15ac

The two factors that have a sum of 19 are 4 and 15.

Note: Because 60 has so many factors, we did not list all of them. b is positive, so we only need to list the

positive factors and stop when we find the two that add up to 19.

Step Two: Split the middle term (bx) into two terms px qx . 26 4 15 10x x x


26 4 15 10

2 3 2 5 3 2

3 2 2 5

x x x

x x x

x x


You Try: Factor the following trinomials.

1. 2 6 5m m 2.

26 2 8y y 3. 221 8 4x x

QOD: Are all quadratic trinomials factorable? If not, write a trinomial that cannot be factored.

Sample CCSD Common Exam Practice Question(s):

Which of the following is a factor of 2

5 23 12x x ?

A. 5 2x

B. 5 3x

C. 5 4x

D. 5 6x



Sample Nevada High School Proficiency Exam Questions (taken from 2009 released version H):

Factor: 210 24x x

A 6 4x x

B 4 6x x

C 2 12x x

D 12 2x x



Skill: factor polynomials connecting the arithmetic and algebraic processes looking for special

products.

Recall: Sum and Difference Pattern 2 2a b a b a b

Square of a Binomial Pattern:

2 2 2

2 2 2

2

2

a b a ab b

a b a ab b

Previously we learned to multiply using the special products patterns. Now we will factor from these special

products.

Special Factoring Patterns

Difference of Two Perfect Squares: 2 2a b a b a b

Perfect Square Trinomial:

22 2

22 2

2

2

a ab b a b

a ab b a b

Ex: Factor 2 9x .

Recognize this as the difference of two perfect squares. 2 22 9 3x x

Factor using the special factoring pattern. 3 3x x


Ex: Factor 216 25y .

Recognize this as the difference of two perfect squares. 2 2216 25 4 5y y

Factor using the special factoring pattern. 4 5 4 5y y


Ex: Factor 2 6 9y y .

Recognize this as a perfect square trinomial. 2 22 6 9 2 3 3y y y y

Factor using the special factoring pattern. 2

3y




Recognize this as a perfect square trinomial. 2 229 12 4 3 2 3 2 2x x x x


3 2x

Ex: Factor 29

16f .

Recognize this as the difference of two perfect squares. 2

22 29 9 3

16 16 4f f f

Factor using the special factoring pattern. 3 3

4 4f f

Ex: Factor 2 24 20 25y xy x .

Recognize this as a perfect square trinomial. 2 22 24 20 25 2 2 2 5 5y xy x y y x x


2 5y x

You Try: Factor the following. 1. 2 249 225n m 2.

249 56 16x x

QOD: Describe how to check your answers when factoring.



Skill: factor polynomials completely.

Factoring Completely: putting it all together

Step One: Factor the GCF using the distributive property. **IMPORTANT!**

Step Two: Factor the polynomial that remains using the ac method or special products.

Ex: Factor completely. 24 36x

Step One: Factor the GCF, which is 4. 24 9x

Step Two: The remaining polynomial is a difference of two perfect squares. Use the special factoring pattern.

4 3 3x x

Ex: Factor completely. 3 22 10 12n n n

Step One: Factor the GCF, which is 2n. 22 5 6n n n

Step Two: Factor the remaining trinomial. 2 6 1n n n

Ex: Factor 4 3 230 58 24y y y .

Step One: Factor the GCF, which is 22y . 2 22 15 29 12y y y

Step Two: Factor the remaining trinomial. 22 5 3 3 4y y y

Note: Because of the large numbers for a and c in the trinomial, a trial and error approach may work best for

factoring.

Ex: Factor 3 23 15 12 60x x x completely.

Step One: Factor the GCF, which is 3. 3 23 5 4 20x x x

Step Two: Factor the remaining polynomial by grouping.

3 2

2

2

3 5 4 20

3 5 4 5

3 5 4

x x x

x x x

x x

Step Three: The underlined factor is a difference of two perfect squares. Use the special factoring pattern to

factor completely.



3 5 2 2x x x

Ex: Factor 5 4 32 32 128x x x .

Step One: Factor the GCF, which is 32x . 3 22 16 64x x x

Step Two: The remaining polynomial is a perfect square trinomial. Use the special factoring pattern.

232 8x x

Quadratic Trinomials: Factorable or Not Factorable?

Discriminant: In the quadratic trinomial 2ax bx c , the discriminant is equal to

2 4b ac

If the trinomial can be factored, the discriminant must be a perfect square.

Ex: Is the trinomial factorable? If yes, factor it. 25 11 2x x

Find the discriminant. 5, 11, 2a b c 22 4 11 4 5 2 121 40 81b ac

81 is a perfect square, so the trinomial is factorable.

Use the ac method. 10 1 10ac

25 1 10 2

5 1 2 5 1

5 1 2

x x x

x x x

x x

Ex: Is the trinomial factorable? If yes, factor it. 23 5 1r r

Find the discriminant. 3, 5, 1a b c 22 4 5 4 3 1 25 12 37b ac

37 is not a perfect square, so the trinomial is not factorable.

Application Problem

Ex: The volume of a box can be modeled by the polynomial 3 23 4 12x x x . Write three linear

binomials that could represent the length, width, and height of the box.

The volume of the box was found by multiplying the length, width, and height. To find the binomials, we will

factor the expression that represents the volume.



Factor by grouping.

3 2

2

2

3 4 12

3 4 3

3 4

x x x

x x x

x x

The underlined factor is a difference of two perfect squares. Use the special pattern to factor.

3 2 2x x x

The length, width, and height of the box are 3, 2, and 2x x x

You Try: List all of the factors of 3 224 6 9b b b .

QOD: Why is it important to factor the GCF as the first step in factoring a polynomial?

Sample CCSD Common Exam Practice Question(s):

The area of a rectangular tabletop is represented by 25 24x x . Which pair of expressions

could represent the dimensions of the tabletop?

A. 6x , 4x

B. 8x , 3x

C. 12x , 2x

D. 24x , 1x

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Algebra I Notes Unit 07b: Polynomials, Factoring and...

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