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Accelerated CCGPS…Math II Name ANSWERS .

Factoring: Difference of Squares & Trinomials Period_____Date_______________

In factoring polynomials, remember that we always look first for a Greatest Common Factor (GCF). That is, the factor common to each term with the largest possible coefficient and the variable(s) to the largest power – no matter how many terms are in the polynomial.

Examples of Greatest Common Factors:

• 20x 4 – 20x 3 + 4x 2 – 4 = 4(5x 4) – 4(5x 3) + 4(x 2) – 4(1) = 4(5x 4 - 5x 3 + x 2 – 1)

• 6m 3n 3 + 3m 3n 2 + m 2n 2 = m 2n 2 (6mn) + m 2n 2 (3m ) + m 2n 2 (1) =

m 2n 2 (6mn + 3m + 1)

Factor the following polynomials by finding the Greatest Common Factor.

1.) 17x 5 + 34x 3 + 51x 2.) 2x 7 – 2x 6 – 64x 5 + 4x 3 3.) 6e 3f - 11ef

17x(x 4 + 2x 2 + 3) 2x 3(x 4 – x 3 – 32x 2 + 2) ef(6e 2 – 11)

4.) 5p 3q 2 + 10p 2q 2 – 20pq 2 5.) 2x 3y – 4x 2y + x 2 6.) x 9 – x 7 + x 4 + x 3

5pq 2(p 2 + 2p – 4) x 2(2xy – 4y + 1) x 3(x 6 – x 4 + x + 1)

Examples of Difference of Squares factoring, etc.:

• x 4 – 81 = (x 2 + 9) (x 2 – 9) = (x 2 + 9) (x + 3) (x – 3)

• 4y 2 - 64 = 4(y 2 – 16) = 4(y + 4 ) (y – 4)

Factor each of the following polynomials completely, using a difference of squares and/or Greatest Common Factor. If the polynomial cannot be factored, write prime.

7.) 32x 2 – 18y 2 8.) 100x 4 - 169 9.) 121y 2 – x 2

= 2(16x2 – 9y2) =

2(4x + 3y)(4x – 3y) (10x 2 + 13)(10x 2 – 13) (11y + x)(11y – x)

10.) 6ax 2 - 48a 11.) x 3y – 25xy 3 12.) 5t 2 – [pic]

= xy(x2 – 25y2) = = 5(t2 – ¼) =

6a(x 2 – 8) xy(x + 5y)(x – 5y) 5(t + ½)(t – ½)

13.) n 8 - 1 14.) -54a 4 + 24b 2 15.) (3x + 4)2 - 49

= (n4 + 1)(n4 – 1)=(n4 + 1)(n2 + 1)(n2 – 1) = = -6(9a4 – 4b2) = = ((3x+4) + 7)((3x+4) – 7) =

(n4 + 1)(n2 + 1)(n +1)(n -1) -6(3a2 + 2b)(3a2 – 2b) (3x + 11)(3x – 3)

To factor a trinomial like x 2 + 7x + 10 in general, think of FOIL in reverse. The first term, x 2, is the result of x times x. Thus the first term of each binomial factor is x :

(x + __) (x + __)

The coefficient of the middle term and the last term of the trinomial are two numbers whose product is 10 and whose sum is 7. Those numbers are 2 and 5. Thus, the factorization is: (x + 2) (x + 5)

Examples of factoring trinomials:

• x 2 + 6x + 5 = (x + 1) (x + 5)

• y 2 - 6y + 9 = (y - 3) (y – 3)

• x 2 + 2x - 8 = (x + 4) (x - 2)

Factor each of the following polynomials completely.

16.) x 2 + 8x + 15 17.) n 2 + 9n + 8 18.) x 2 – 9x + 14

(x + 5)(x + 3) (n + 8)(n + 1) (x – 7)(x - 2)

19.) x 2 – 11x + 28 20.) a 2 + 10a + 21 21.) x 2 – 12x + 36

(x – 7)(x – 4) (a + 7)(a + 3) (x – 6)(x – 6)

22.) n 2 – 2n - 15 23.) x 2 – 7x - 18 24.) x 2 + 2x – 99

(n + 3)(n – 5) (x + 2)(x – 9) (x + 11)(x – 9)

25.) x 2 – 4x + 4 26.) x 2 – 14x + 48 27.) x 2 – 14x + 45

(x – 2)(x – 2) (x – 8)(x – 6) (x – 9)(x – 5)

28.) x 2 – 10x + 24 29.) y 2 + y - 42 30.) x 2 – 6x - 72

(x – 6)(x – 4) (y + 7)(y – 6) (x - 12)(x + 6)

31.) x 2 + 11x + 30 32.) x 2 – 14x + 49 33.) x 2 + 29x + 100

(x + 6)(x + 5) (x – 7)(x – 7) (x + 25)(x + 4)

34.) x 2 + 20x + 100 35.) x 2 - 21x - 100 36.) x 3 + 16x 2 + 64x

(Careful with this one!)

(x + 10)(x + 10) (x + 4)(x – 25) x(x + 8)(x + 8)

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Important thought: How do you know what signs to use in the binomial factors? Do you see a pattern with the addition or subtraction?