# Multiplying Polynomials Distribute and FOIL Polynomials Polynomials Multiplying

Multiplying Polynomials Distribute and FOIL

Polynomials * Polynomials Multiplying a Polynomial by another Polynomial requires more than one distributing step. Multiply: (2 a + 7 b)(3 a + 5 b) Distribute 2 a(3 a + 5 b) and distribute 7 b(3 a + 5 b): 6 a 2 + 10 ab 21 ab + 35 b 2 Then add those products, adding like terms: 6 a 2 + 10 ab + 21 ab + 35 b 2 = 6 a 2 + 31 ab + 35 b 2

Polynomials * Polynomials An alternative is to stack the polynomials and do long multiplication. (2 a + 7 b)(3 a + 5 b) (2 a + 7 b) x (3 a + 5 b) Multiply by 5 b, then by 3 a: When multiplying by 3 a, line up the first term under 3 a. Add like terms: (2 a + 7 b) x (3 a + 5 b) 21 ab + 35 b 2 + 6 a 2 + 10 ab 6 a 2 + 31 ab + 35 b 2

Polynomials * Polynomials Multiply the following polynomials:

Polynomials * Polynomials (x + 5) x (2 x + -1) -x + -5 + 2 x 2 + 10 x 2 x 2 + 9 x + -5 (3 w + -2) x (2 w + -5) -15 w + 10 + 6 w 2 + -4 w 6 w 2 + -19 w + 10

Polynomials * Polynomials (2 a 2 + a + -1) x (2 a 2 + 1) 2 a 2 + a + -1 + 4 a 4 + 2 a 3 + -2 a 2 4 a 4 + 2 a 3 + a + -1

Types of Polynomials • We have names to classify polynomials based on how many terms they have: Monomial: a polynomial with one term Binomial: a polynomial with two terms Trinomial: a polynomial with three terms

F. O. I. L. There is an acronym to help us remember how to multiply two binomials without stacking them. (2 x + -3)(4 x + 5) F : Multiply the First term in each binomial. 2 x • 4 x = 8 x 2 O : Multiply the Outer terms in the binomials. 2 x • 5 = 10 x I : Multiply the Inner terms in the binomials. -3 • 4 x = -12 x L : Multiply the Last term in each binomial. -3 • 5 = -15 (2 x + -3)(4 x + 5) = 8 x 2 + 10 x + -12 x + -15 = 8 x 2 + -2 x + -15

F. O. I. L. Use the FOIL method to multiply these binomials: 1) (3 a + 4)(2 a + 1) 2) (x + 4)(x - 5) 3) (x + 5)(x - 5) 4) (c - 3)(2 c - 5) 5) (2 w + 3)(2 w - 3)

F. O. I. L. Use the FOIL method to multiply these binomials: 1) (3 a + 4)(2 a + 1) = 6 a 2 + 3 a + 8 a + 4 = 6 a 2 + 11 a + 4 2) (x + 4)(x - 5) = x 2 + -5 x + 4 x + -20 = x 2 + -1 x + -20 3) (x + 5)(x - 5) = x 2 + -5 x + -25 = x 2 + -25 4) (c - 3)(2 c - 5) = 2 c 2 + -5 c + -6 c + 15 = 2 c 2 + -11 c + 15 5) (2 w + 3)(2 w - 3) = 4 w 2 + -6 w + -9 = 4 w 2 + -9

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