Warm-Up #22 (3m + 4)(-m + 2) Factor out ( 96𝑥 3 − 48𝑥 2 +60𝑥)

Factoring Binomials

Definitions To factor an expression means to write an equivalent expression that is a product To factor a polynomial means to write the polynomial as a product of other polynomials A factor that cannot be factored further is said to be a prime factor (prime polynomial) A polynomial is factored completely if it is written as a product of prime polynomials

Distributive Property using FOIL A(B + C) = AB + AC A(B – C) = AB – AC Example: 5x(6x + 3)

Distributive Property Example: (x+2)(6x + 3)

Distributive Property Example: 5x(6x + 3)(-2x – 1)

Factor the expression 30𝑥 2 +15𝑥 5x(6x + 3)

Ex: Factor 6a5 – 3a3 – 2a2 Recall: GCF(6a5,3a3,2a2)= a2 1 6a5 – 3a3 – 2a2 = a2( - - ) 6a3 3a 2 a2 a2 a2  6a5 – 3a3 – 2a2 = a2(6a3 – 3a – 2)

Ex: Factor x(a + b) – 2(a + b) Always ask first if there is common factor the terms share . . . x(a + b) – 2(a + b) Each term has factor (a + b)  x(a + b) – 2(a + b) = (a + b)( – ) x 2 (a + b) (a + b)  x(a + b) – 2(a + b) = (a + b)(x – 2)

Ex: Factor a(x – 2) + 2(2 – x) As with the previous example, is there a common factor among the terms? Well, kind of . . . x – 2 is close to 2 - x . . . Hum . . . Recall: (-1)(x – 2) = - x + 2 = 2 – x  a(x – 2) + 2(2 – x) = a(x – 2) + 2((-1)(x – 2)) = a(x – 2) + (– 2)(x – 2) = a(x – 2) – 2(x – 2)  a(x – 2) – 2(x – 2) = (x – 2)( – ) a 2 (x – 2) (x – 2)

Ex: Factor b(a – 7) – 3(7 – a) Common factor among the terms? Well, kind of . . . a – 7 is close to 7 - a Recall: (-1)(a – 7) = - a + 7 = 7 – a  b(a – 7) – 3(7 – a) = b(a – 7) – 3((-1)(a – 7)) = b(a – 7) + 3(a – 7) = b(a – 7) +3(a – 7)  b(a – 7) + 3(a – 7) = (a – 7)( + ) b 3 (a – 7) (a – 7)

Special Binomials Is the polynomial a difference of squares? a2 – b2 = (a – b)(a + b) Is the trinomial a perfect-square trinomial? a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2

Special Binomials Is the polynomial a difference of squares? a2 – b2 = (a – b)(a + b) LIST ALL OF THE PERFECT SQUARES:

Ex: Factor x2 – 4 Notice the terms are both perfect squares and we have a difference  difference of squares x2 = (x)2 4 = (2)2  x2 – 4 = (x)2 – (2)2 = (x – 2)(x + 2) a2 – b2 = (a – b)(a + b) factors as

Example 1: 𝑥 2 −36 Example 2: 𝑥 2 −125

Ex: Factor 9p2 – 16 Notice the terms are both perfect squares and we have a difference  difference of squares 9p2 = (3p)2 16 = (4)2  9a2 – 16 = (3p)2 – (4)2 = (3p – 4)(3p + 4) a2 – b2 = (a – b)(a + b) factors as

Example 1: 16𝑥 2 −25 Example 2: 17𝑥 2 −49

Ex: Factor y6 – 25 Notice the terms are both perfect squares and we have a difference  difference of squares y6 = (y3)2 25 = (5)2  y6 – 25 = (y3)2 – (5)2 = (y3 – 5)(y3 + 5) a2 – b2 = (a – b)(a + b) factors as

Example 1: 𝑚 8 −36 Example 2: 𝑥 6 −64

Ex: Factor 81 – x2y2 Notice the terms are both perfect squares and we have a difference  difference of squares 81 = (9)2 x2y2 = (xy)2  81 – x2y2 = (9)2 – (xy)2 = (9 – xy)(9 + xy) a2 – b2 = (a – b)(a + b) factors as

Example 1: −36 + 𝑟 2 𝑚 2 Example 2: −125 + 𝑛 4 𝑥 2