1. Factoring Quadratic Expressions Lesson 4-4 Part 1
Algebra 2
Factoring Quadratic Expressions
Lesson 4-4 Part 1

2. Goals
Goal
Rubric
To find common binomial factors of quadratic expressions.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
Factoring
Greatest Common Factor (GCF) of an Expression

4. Big Idea: Solving Equations and Inequalities
Essential Question
Big Idea: Solving Equations and Inequalities
How is factoring related to the Distributive Property?

5. Definition
Factoring – rewriting an expression as a product of its factors.
Factoring a polynomial reverses the multiplication process (factoring is unmultiplying).

6. Factors 4 · 9 = 36 3(x + 2) = 3x + 6 Factors
(2x – 7)(3x + 5) = 6x2 – 11x – 35
Factors
The expressions on the left side are called factors of the expression on the right side.
To factor a polynomial means to write the polynomial as a product of two or more polynomials.
The greatest common factor (GCF) of a list of algebraic expressions is the largest expression that divides evenly into all the expressions.

7. Greatest Common Factor
How to Find the Greatest Common Factor of a List of Numbers
Step 1: Write each number as a product of prime factors.
Step 2: Determine the common prime factors.
Step 3: Find the product of the common factors found in Step 2. This number is the GCF.
Example:
Find the GCF of 16 and 20.
16 = 2 · 2 · 2 · 2
The common factors are 2 and 2.
20 = 2 · · 5
The GCF is 2 · 2 = 4.

8. Greatest Common Factor
Example:
Find the GCF of 60, 75, and 135.
60 = 2 · 2 · · 5
The common factors are 3 and 5.
75 = · 5 · 5
135 = · 3 · 3 · 5
The GCF is 3 · 5 = 15.

9. Your Turn: Find the GCF of each list of numbers. 6, 8 and 46 6 = 2 · 3
8 = 2 · 2 · 2
46 = 2 · 23
So the GCF is 2.
144, 256 and 300
144 = 2 · 2 · 2 · 3 · 3
256 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2
300 = 2 · 2 · 3 · 5 · 5
So the GCF is 2 · 2 = 4.

10. The GCF as a Binomial Example:
Find the GCF of 6(x – y) and 15(x – y)3.
6(x – y) = 2 · · (x – y)
15(x – y)3 = · 5 · (x – y) · (x – y) · (x – y)
The GCF is 3 · (x – y) = 3(x – y).

11. Finding the GCF
Steps to Find the Greatest Common Factor of a polynomial
Step 1: Find the GCF of the coefficients of each variable factor.
Step 2: For each variable factor common to all terms, determine the smallest exponent that the variable factor is raised to.
Step 3: Find the product of the common factors found in Steps 1 and 2. This expression is the GCF.

12. Example: Find the GCF of the following list of terms.
a3b2, a2b5 and a4b7
a3b2 = a · a · a · b · b
a2b5 = a · a · b · b · b · b · b
a4b7 = a · a · a · a · b · b · b · b · b · b · b
So the GCF is a · a · b · b = a2b2
Notice that the GCF of terms containing variables will use the smallest exponent found amongst the individual terms for each variable.

13. Example: Find the GCF of each pair of monomials. 15x3 and 9x2
Write the factorization of each coefficient and write powers as products.
15x3 = 3  5  x  x  x
9x2 = 3  3  x  x
Align the common factors.
3  x  x = 3x2
Find the product of the common factors.
The GCF of 3x3 and 6x2 is 3x2.

14. Your Turn: Find the GCF of each list of terms. x3 and x7
x3 = x · x · x
x7 = x · x · x · x · x · x · x
So the GCF is x · x · x = x3
6x5 and 4x3
6x5 = 2 · 3 · x · x · x
4x3 = 2 · 2 · x · x · x
So the GCF is 2 · x · x · x = 2x3

15. Factoring Polynomials
Steps to Factor a Polynomial Using the GCF
Step 1: Identify the GCF of the terms that make up the polynomial.
Step 2: Rewrite each term as the product of the GCF and the remaining factor.
Step 3: Use the Distributive Property “in reverse” to factor out the GCF.
Step 4: Check using the Distributive Property.

16. Factoring Polynomials
Example: Factor the trinomial 36a6 + 45a4 – 18a2 by factoring out the GCF.
Step 1: Find the GCF.
GCF = 9a2
Step 2: Rewrite each term as the product of the GCF and the remaining term.
36a6 + 45a4 – 18a2 = 9a2 · 4a4 + 9a2 · 5a2 – 9a2 · 2
Step 3: Factor out the GCF.
36a6 + 45a4 – 18a2 = 9a2(4a4 + 5a2 – 2)
Step 4: Check.
9a2(4a4 + 5a2 – 2) = 36a6 + 45a4 – 18a2 

17. Factoring Out a Negative Number
Example: Factor – 3x6 + 9x4 – 18x by factoring out the GCF:
Step 1: Find the GCF.
GCF = – 3x
Step 2: Rewrite each term as the product of the GCF and the remaining term.
– 3x6 + 9x4 – 18x = – 3x · x5 + (– 3x)(– 3x3) + (– 3x) · 6
Step 3: Factor out the GCF.
– 3x6 + 9x4 – 18x = – 3x(x5 – 3x3 + 6)
Step 4: Check.
– 3x(x5 – 3x3 + 6) = – 3x6 + 9x4 – 18x 

18. Your Turn:  Factor each polynomial. Check your answer. 5b + 9b3
Find the GCF.
The GCF of 5b and 9b3 is b.
Write terms as products using the GCF as a factor.
5(b) + 9b2(b)
Use the Distributive Property to factor out the GCF.
b(5 + 9b2)
Multiply to check your answer.
b(5 + 9b2)
Check
The product is the original polynomial.
5b + 9b3 

19. Your Turn: Factor each polynomial. 9d2 – 82 9d2 = 3  3  d  d
Find the GCF.
82 =  2  2  2  2  2
There are no common factors other than 1.
9d2 – 82
The polynomial cannot be factored further.

20. Your Turn:  Factor each polynomial. Check your answer.
8x3 – 4x2 – 16x
Find the GCF.
The GCF of 8x3, 4x2, and 16x is 4x.
Write terms as products using the GCF as a factor.
2x2(4x) – x(4x) – 4(4x)
4x(2x2 – x – 4)
Use the Distributive Property to factor out the GCF.
Check
4x(2x2 – x – 4)
Multiply to check your answer.
The product is the original polynomials.
8x3 – 4x2 – 16x 

21. Your Turn: Factor each polynomial. 8x4 + 4x3 – 2x2
Find the GCF.
The GCF of 8x4, 4x3 and –2x2 is 2x2.
Write terms as products using the GCF as a factor.
4x2(2x2) + 2x(2x2) –1(2x2)
2x2(4x2 + 2x – 1)
Use the Distributive Property to factor out the GCF.
Check 2x2(4x x – 1)
Multiply to check your answer.
8x4 + 4x3 – 2x2
The product is the original polynomial.

22. Example: Factoring the GCF
Factor each polynomial.
–14x – 12x2
Find the GCF.
The GCF of 14x and 12x2 is -2x.
7(-2x) + 6x(-2x)
Write each term as a product using the GCF.
–2x( x)
Use the Distributive Property to factor out the GCF.

23. Factoring Out a Binomial
Example: Factor out the greatest common binomial factor: (3x + y) – z(3x + y)
GCF = 3x + y
6(3x + y) – z(3x + y) = (3x + y)(6 – z)
Check:
(3x + y)(6 – z) = 6(3x + y) – z(3x + y) 

24. Your Turn: Factor out the GCF in each of the following polynomials.
1) 6(x + 2) – y(x + 2) =
6 · (x + 2) – y · (x + 2) =
(x + 2)(6 – y)
2) xy(y + 1) – (y + 1) =
xy · (y + 1) – 1 · (y + 1) =
(y + 1)(xy – 1)

25. Factoring the GCF
Remember that factoring out the GCF from the terms of a polynomial should always be the first step in factoring a polynomial.
This will usually be followed by additional steps in the process.

26. Quadratic Trinomials
A quadratic trinomial is a polynomial of the form ax2 + bx + c, a  0 where a represents the coefficient of the squared (second degree) term, b represents the coefficient of the linear (first degree) term and c represents the constant.
3x2 + 4x + 7
8a2 – 24a – 10
6c2 + c – 25
leading coefficient
When the trinomial is written in standard form (or descending order of degree), the coefficient of the squared term is called the leading coefficient.

27. Factoring a Trinomial x2 + bx + c
In this lesson, you will learn how to factor a trinomial into two binominals.
Notice that when you multiply (x + 2)(x + 5), the constant term in the trinomial is the product of the constants in the binomials.
(x + 2)(x + 5) = x2 + 7x + 10

28. Factoring a Trinomial x2 + bx + c
To factor a simple trinomial of the form x 2 + bx + c (leading coefficient is 1), express the trinomial as the product of two binomials. For example,
x x + 24 = (x + 4)(x + 6).
Factoring these trinomials is based on reversing the FOIL process.

29. Factoring a Trinomial x2 + bx + c
Look at the product of (x + a) and (x + b). x2 ab
(x + a)(x + b) = x2 + ax + bx + ab ax
= x2 + (a + b)x + ab bx
The coefficient of the middle term is the sum of a and b. The constant term is the product of a and b.

30. Factoring a Trinomial x2 + bx + c
Recall by using the FOIL method that
F O I L
(x + 2)(x + 4) = x 2 + 4x + 2x + 8
= x 2 + 6x + 8
To factor x 2 + bx + c into (x + one #)(x + another #), note that b is the sum of the two numbers and c is the product of the two numbers.
So we’ll be looking for 2 numbers whose product is c and whose sum is b.
Note: there are fewer choices for the product, so that’s why we start there first.

31. Factoring Factoring a Trinomial of the Form x2 + bx + c
Step 1: Find the pair of integers whose product is c and whose sum is b. That is, determine m and n such that mn = c and m + n = b.
Step 2: Write x2 + bx + c = (x + m)(x + n).
Step 3: Check your work by multiplying the binomials.
The coefficient of x is the sum of the two numbers.
x2 – 11x = (x – 3)(x – 8)
The last term is the product of the two numbers.

32. Example: Factor: x2 + 8x + 15 x2 + 8x + 15 = (x + ?)(x + ?)
___ × ___ = 15 3 5
Find two numbers that we can multiply together to get 15 and add together to get 8.
___ + ___ = 8 3 5
x2 + 8x + 15 = (x + 3)(x + 5)
Check:
(x + 3)(x + 5) = x2 + 8x + 15 

33. Factoring Trinomials x2 + bx + c TIP
When c is positive, its factors have the same sign. The sign of b tells you whether the factors are positive or negative. When b is positive, the factors are positive and when b is negative, the factors are negative.

34. Example: c is Positive   (x + )(x + )
Factor each trinomial. Check your answer.
x2 + 6x + 5
(x + )(x + )
b = 6 and c = 5; look for factors of 5 whose sum is 6.
Factors of 5 Sum
1 and 
The factors needed are 1 and 5.
(x + 1)(x + 5)
Check (x + 1)(x + 5) = x2 + 5x + x + 5
Use the FOIL method.
= x2 + 6x + 5 
The product is the original trinomial.

35. Example: c is Positive (x + )(x + )   
Factor each trinomial. Check your answer.
x2 + 6x + 9
(x + )(x + )
b = 6 and c = 9; look for factors of 9 whose sum is 6.
Factors of 9 Sum
1 and 
3 and 
The factors needed are 3 and 3.
(x + 3)(x + 3)
Check (x + 3)(x + 3 ) = x2 + 3x + 3x + 9
Use the FOIL method.
= x2 + 6x + 9 
The product is the original trinomial.

36. Example: c is Positive (x + )(x + )   
Factor each trinomial. Check your answer.
x2 – 8x + 15
(x + )(x + )
b = –8 and c = 15; look for factors of 15 whose sum is –8.
Factors of –15 Sum
–1 and –15 –16 
–3 and –5 –8 
The factors needed are –3 and –5 .
(x – 3)(x – 5)
Check (x – 3)(x – 5 ) = x2 – 5x – 3x + 15
Use the FOIL method.
= x2 – 8x + 15 
The product is the original trinomial.

37. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 + 8x + 12
(x + )(x + )
b = 8 and c = 12; look for factors of 12 whose sum is 8.
Factors of Sum
1 and 
2 and 
The factors needed are 2 and 6 .
(x + 2)(x + 6)
Check (x + 2)(x + 6 ) = x2 + 6x + 2x + 12
Use the FOIL method.
= x2 + 8x + 12 
The product is the original trinomial.

38. Your Turn: (x + )(x+ )    Factor each trinomial. Check your answer.
b = –5 and c = 6; look for factors of 6 whose sum is –5.
Factors of 6 Sum
–1 and –6 –7 
–2 and –3 –5 
The factors needed are –2 and –3.
(x – 2)(x – 3)
Check (x – 2)(x – 3) = x2 – 3x – 2x + 6
Use the FOIL method.
= x2 – 5x + 6 
The product is the original trinomial.

39. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 + 13x + 42
(x + )(x + )
b = 13 and c = 42; look for factors of 42 whose sum is 13.
Factors of 42 Sum
1 and 
6 and 7 13 
2 and
The factors needed are 6 and 7.
(x + 6)(x + 7)
Check (x + 6)(x + 7) = x2 + 7x + 6x + 42
Use the FOIL method.
= x2 + 13x + 42 
The product is the original trinomial.

40. Your Turn: (x + )(x+ )    Factor each trinomial. Check your answer.
b = –13 and c = 40; look for factors of 40 whose sum is –13.
(x + )(x+ )
Factors of 40 Sum 
–2 and –20 –22
–4 and –10 –14 
–5 and – –13
The factors needed are –5 and –8.
(x – 5)(x – 8)
Check (x – 5)(x – 8) = x2 – 8x – 5x + 40
Use the FOIL method.
= x2 – 13x + 40 
The product is the original polynomial.

41. Factoring Trinomials x2 + bx + c TIP
When c is negative, its factors have opposite signs. The sign of b tells you which factor is positive and which is negative. The factor with the greater absolute value has the same sign as b.

42. Example: c is Negative (x + )(x + )   Factor each trinomial.
b = 1 and c = –20; look for factors of –20 whose sum is 1. The factor with the greater absolute value is positive.
Factors of –20 Sum 
–1 and
–2 and 
–4 and
The factors needed are 5 and –4.
(x – 4)(x + 5)

43. Example: c is Negative (x + )(x + )   Factor each trinomial.
b = –3 and c = –18; look for factors of –18 whose sum is –3. The factor with the greater absolute value is negative.
Factors of –18 Sum 
1 and – –17
2 and – – 7 
3 and – – 3
The factors needed are 3 and –6.
(x – 6)(x + 3)

44. If you have trouble remembering the rules for which factor is positive and which is negative, you can try all the factor pairs and check their sums.
Helpful Hint

45. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 + 2x – 15
b = 2 and c = –15; look for factors of –15 whose sum is 2. The factor with the greater absolute value is positive.
(x + )(x + )
Factors of –15 Sum 
–1 and
–3 and 
The factors needed are –3 and 5.
(x – 3)(x + 5)
Check (x – 3)(x + 5) = x2 + 5x – 3x – 15
Use the FOIL method.
= x2 + 2x – 15 
The product is the original polynomial.

46. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 – 6x + 8
(x + )(x + )
b = –6 and c = 8; look for factors of 8 whose sum is –6.
Factors of 8 Sum 
–1 and – –7
–2 and – –6 
The factors needed are –4 and –2.
(x – 2)(x – 4)
Check (x – 2)(x – 4) = x2 – 4x – 2x + 8
Use the FOIL method.
= x2 – 6x + 8 
The product is the original polynomial.

47. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 – 8x – 20
(x + )(x + )
b = –8 and c = –20; look for factors of –20 whose sum is –8. The factor with the greater absolute value is negative.
Factors of –20 Sum 
1 and – –19
2 and – –8 
(x – 10)(x + 2)
The factors needed are –10 and 2.
Check (x – 10)(x + 2) = x2 + 2x – 10x – 20
Use the FOIL method.
= x2 – 8x – 20 
The product is the original polynomial.

48. Trinomials of the Form x2 + bx + c
The following table summarizes the four forms for factoring a quadratic trinomial in the form x2 + bx + c.
Form
Signs of m and n
Example
x2 + bx + c, where b and c are both positive
m and n are both positive
x2 + 3x + 2 = (x + 2)(x + 1)
x2 + bx + c, where b is negative and c is positive
m and n are both negative
a2 – 7a + 12 = (a – 4)(a – 3)
x2 + bx + c, where b is positive and c is negative
m and n are opposite in sign and the factor with the larger absolute value is positive
y2 + 2y – 24 = (y + 6)(y – 4)
x2 + bx + c, where b and c are both negative
m and n are opposite in sign and the factor with the larger absolute value is negative
b2 – 4b – 21 = (b – 7)(b + 3)

49. Practice

50. More Practice

51. Example: Prime Polynomials
A polynomial that cannot be written as the product of two other polynomials (other than 1 or – 1) is said to be a prime polynomial.
Example: Factor: 5x2  x  2
The Factors of 5 are:
The Factors of 2 are:
1 and 5
1 and 2
Possible Factors:
Middle Term:
Neither of these sums produce the correct middle term.
(x – 1)(5x + 2)
– 3x
(x – 2)(5x + 1)
– 9x
The polynomial 5x2  x  2 is prime.

52. Your Turn: Factor the polynomial x 2 – 6x + 10.
Since our two numbers must have a product of 10 and a sum of – 6, the two numbers will have to both be negative.
Negative factors of 10 Sum of Factors
– 1, – – 11
– 2, – – 7
Since there is not a factor pair whose sum is – 6,
x 2 – 6x +10 is not factorable and we call it a prime polynomial.

53. VERY IMPORTANT
When factoring, always factor out the GCF first.

54. Example: Trinomials with a GCF
Factor: 2x2 – 32x + 96
2x2 – 32x + 96
= 2(x2 – 16x + 48)
The common factor of 2 can be factored out.
x2 – 16x + 48
= (x – ?)(x – ?)
Find two numbers that we can multiply together to get 48 and add together to get – 16.
( 12)
____ × ____ = 48
( 4)
( 12)
( 4)
____ + ____ = – 16
2x2 – 32x + 96 = 2(x – 12)(x  4)
Check:
2(x – 12)(x  4) = 2(x2 – 16x + 48)
= 2x2 – 32x + 96 

55. Your Turn: Factor: 2x2 + 4x – 30 Factor: –3x2 + 9x – 6
Factor: 5x3 – 5x2 – 30x
Solution: 2(x + 5)(x – 3)
Solution: –3(x – 1)(x – 2)
Solution: 5x(x + 2)(x – 3)

56. Example: Negative Leading Coefficients
Factor: – x2 – 12x – 36
It is easier to factor out a GCF of – 1.
– x2 – 12x – 36 = – 1(x2 + 12x + 36)
– 1(x2 + 12x + 36) = – 1(x + ?)(x + ?)
___ × ___ = 36 6 6
Find two numbers that we can multiply together to get 36 and add together to get 12.
___ + ___ = 12 6 6
– 1(x2 + 12x + 36) = – 1(x + 6)(x + 6)
= –(x + 6)2
Check:
–(x + 6)2 = – 1(x + 6)(x + 6) 
= – x2 – 12x – 36

57. Your Turn: Factor: -x2 + 7x + 18 = -1(x2 - 7x - 18)
= -(x – 9)(x + 2) or (9 - x)(x + 2)

58. Big Idea: Solving Equations and Inequalities
Essential Question
Big Idea: Solving Equations and Inequalities
How is factoring related to the Distributive Property?
Factoring is rewriting an expression as a product of its factors. This process is the reverse of the Distributive Property. Use your knowledge of the Distributive Property to write a quadratic expression as a product.

59. Assignment
Section 4-4 part 1, Pg 229 – 230; #1 – 8 all, 10 – 38 even