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.
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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.
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
Alg I Unit 07b Notes PolynomialsFactorSpecial Products Page 2 of 18 3/18/2013
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
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
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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
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
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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
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
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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
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
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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
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
Alg I Unit 07b Notes PolynomialsFactorSpecial Products Page 7 of 18 3/18/2013
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
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
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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
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
Alg I Unit 07b Notes PolynomialsFactorSpecial Products Page 9 of 18 3/18/2013
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?
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
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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!
Ex: Factor 2 10 24x x .
Step One: Find values for p and q such that 24pq and 10p q .
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 .
Step Two: Write each factor x p x q 4 6x x
Check your answer using FOIL!
Ex: Factor 2 8 9x x .
Step One: Find values for p and q such that 9pq and 8p q .
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 .
Step Two: Write each factor x p x q 9 1x x
Check your answer using FOIL!
Algebra I Notes Unit 07b: Polynomials, Factoring and Special Products Unit 07b
Alg I Unit 07b Notes PolynomialsFactorSpecial Products Page 11 of 18 3/18/2013
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
Check your answer using FOIL!
Ex: Factor 26 11 2n n .
Step One: Multiply ac. Find values for p and q such that pq ac , and p q b
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
Step Three: Factor by grouping.
26 1 12 2
6 1 2 6 1
6 1 2
n n n
n n n
n n
Check your answer using FOIL!
Ex: Factor 24 9 5t t .
Step One: Multiply ac. Find values for p and q such that pq ac , and p q b
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