Lesson 83 Factoring special products Look for a

Lesson 83 Factoring special products

Look for a pattern in the products n (x+1)2 = (x+1)= x 2+2 x+1 x 2 +2(1 x)+12 n (x+2)2= (x+2)= x 2 +4 x+4 x 2 +2(2 x) +22 n (x+3)2=(x+3)= x 2+6 x+9 x 2 +2(3 x)+32 n (x-1)2=(x-1) = x 2 -2 x + 1 x 2 +2(-1 x)+(-1)2 n (x-2)2 = (x-2)= x 2 - 4 x +4 x 2 +2(-2 x) +(-2)2 n Square the first term n Square the 2 nd term n Multiply the product of both terms by 2

Perfect square trinomials n the factored form of a perfect square trinomial is: 2 2 2 n a + 2 ab + b = (a+b) n a 2 -2 ab + b 2 = (a-b)2

Factoring perfect square trinomials n Determine if each polynomial is a perfect square trinomial. If it is, factor it! n x 2 + 6 x + 9 n write in perfect square trinomial form n x 2 +2(3 x) + 32 n =(x+3)2 n x 2 -2 x + 4 n x 2 +2(-1 x) +22 not perfect square form

practice n 36 x 2 -48 x+16 n Factor GCF first n 4(9 x 2 -12 x+4) n Write in perfect square form n 4 ( (3 x)2 +2(-3 x)(2) + 22 ) n =4(3 x-2)2

Difference of 2 squares 2 n a - 2 b n example: x 2 -49 n x 2 -(7)2 n (x-7)(x+7) n n = (a+b)(a-b)

practice n Are these the difference of 2 squares. If so, factor them. n 4 x 2 - 25 n 9 m 2 -16 n 6 n X 2 -8 n 100 x 2 - 25 n 2 x 6 -288 n X 2 + 9 n -36 + x 10 n -64+z 8

Lesson 87 n. Factoring by grouping

Polynomials can be factored by grouping n When a polynomial has 4 terms, make 2 groups and factor out the GCF from each group. n 2 x 2 + 4 xy + 7 x + 14 y n (2 x 2 + 4 xy) + (7 x + 14 y) n 2 x(x+2 y) +7 (x + 2 y) n Now factor out the common factor from each binomial n (x+2 y)(2 x+7)

Factor by grouping n 2 5 x + 10 xy + 3 x + 6 y

Rearranging before grouping n Sometimes it is necessary to rearrange the terms so that there are common factors n 3 y 2 - 8 y 3 - 8 y + 3 n Maybe it would be better to rearrange n 3 y 2 + 3 -8 y n Then 3(y 2+1) -8 y(y 2+1) n (y 2+1)(3 -8 y)

practice 2 n 5 x - 3 12 x - 12 x + 5

Factoring with the GCF n Always factor out GCF first n 45 a 3 b - 15 a 3 + 15 a 2 b - 5 a 2 n GCF is 5 a 2 n 5 a 2(9 ab-3 a +3 b -1) n Group n 5 a 2(9 ab-3 a) +(3 b-1) n 5 a 2(3 a(3 b-1) +(3 b-1)) n 5 a 2(3 b-1)(3 a+1)

practice 3 n 54 xz 2 +6 xz - 3 18 z - 2 2 z

Factoring with opposites n factor 3 a 2 b - 18 a + 30 -5 ab n Group n (3 a 2 b-18 a) + (30 -5 ab) n 3 a(ab-6) + 5(6 -ab) n Factor out -1 from 2 nd group n 3 a(ab-6) + 5(-1)(-6+ab) n 3 a(ab-6) -5(ab-6) n (ab-6)(3 a-5)

practice 2 n 5 x y - 3 xy - 50 x + 30

Factoring a trinomial by grouping n n n ax 2 + bx + c 1) find the product of ac 2) find 2 factors of ac with a sum equal to b 3) write the trinomial using the sum 4) factor by grouping Example: 2 x 2 + 11 x + 15 2 x 15 = 30 Find factors of 30 that add up to 11 6, 5 2 x 2 + 5 x + 6 x + 15 Factor by grouping

factor 2 nx - 7 x - 44 2 n 6 k - 17 k + 10 nx 2 - 4 x - 21 n 5 k 2 - 13 k + 6

Investigation 9 n. Choosing a factoring method

Checklist for factoring n 1) look for the GCF- does each term have a common n n factor? 2)look for a difference of 2 squares- are there only 2 terms of the polynomial and are they subtracted? 3)look for perfect square trinomials- are the first and last terms perfect squares and is the second term the product of the square roots of the first and last terms? 4) are there 3 terms in the polynomial and are they all being added-is the last term not a perfect square? 5)are there 4 terms in the polynomial- if they have no GCF can you group them into 2 groups that hace common factors?

factor n x 2 + 2 x +1 n 3 x 2 + xy - 12 x - 4 y n 3 x 2 + 13 x + 4 n x 2 + 9 x + 20 n 2 x 2 + 8 x + 6 n 5 x 4 - 5 x 2 n 9 x 2 + 30 x + 25 n x 2 -9