9-2 Multiplying and Factoring *Using the Distributive Property

To simplify a product of monomials (4x)(2x) Use the Commutative and Associative Properties of Multiplication to group the numerical coefficients and to group like variable; Find the product of the numbers Use the properties of exponents to simplify the variable product. So your answer is (4x)(2x) = 8x2 (4x)(2x) = (4 · 2)(x · x ) = (4 · 2) = 8 (x · x) = x1 · x1 = x1+1 = x2

Multiply powers with the same base: You can also use the Distributive Property for multiplying powers with the same base when multiplying a polynomial by a monomial. Simplify -4y2(5y4 – 3y2 + 2) -4y2(5y4 – 3y2 + 2) = -4y2(5y4) – 4y2(-3y2) – 4y2(2) = Use the Distributive Property -20y2 + 4 + 12y2 + 2 – 8y2 = Multiply the coefficients and add the -20y6 + 12y4 – 8y2 exponents of powers with the same base. Remember, Multiply powers with the same base: 35 · 34 = 35 + 4 = 39

Multiplying powers with the same base. Simplify each product. a) 4b(5b2 + b + 6) b) -7h(3h2 – 8h – 1) c) 2x(x2 – 6x + 5) d) 4y2(9y3 + 8y2 – 11) Remember, Multiplying powers with the same base. 20b3 + 4b2 + 24b -21h3 + 56h2 + 7h 2x3 -12x2 + 10x 36y5 + 32y4 – 44y2

Factoring a Monomial from a Polynomial Factoring a polynomial reverses the multiplication process. To factor a monomial from a polynomial, first find the greatest common factor (GCF) of its terms. The GCF is what the terms all have in common! Find the GCF of the terms of: 4x3 + 12x2 – 8x List the prime factors of each term. 4x3 = 2 · 2 · x · x x 12x2 = 2 · 2 · 3 · x · x 8x = 2 · 2 · 2 · x The GCF is 2 · 2 · x or 4x.

Find the GCF of the terms of each polynomial Find the GCF of the terms of each polynomial. a) 5v5 + 10v3 b) 3t2 – 18 c) 4b3 – 2b2 – 6b d) 2x4 + 10x2 – 6x 5v3 3 2b 2x

Factoring Out a Monomial To factor a polynomial completely, you must factor until there are no common factors other than 1. Factor 3x3 – 12x2 + 15x Step 1 Find the GCF 3x3 = 3 · x · x · x 12x2 = 2 · 2 · 3 · x · x 15x = 3 · 5 · x The GCF is 3 · x or 3x Step 2 Factor out the GCF 3x3 – 12x2 + 15x = 3x(x2) + 3x(-4x) + 3x(5) = 3x(x2 – 4x + 5)

Use the GCF to factor each polynomial Use the GCF to factor each polynomial. a) 8x2 – 12x b) 5d3 + 10d c) 6m3 – 12m2 – 24m d) 4x3 – 8x2 + 12x 4x(2x-3) 5d(d2 + 2) 6m(m2 -2m -4) 4x(x2 –2x +3) Try to factor mentally by scanning the coefficients of each term to find the GCF. Next, scan for the least power of the variable.

Pg. 463 2-24, 30-38 Evens only! Due Monday! Homework/Classwork Pg. 463 2-24, 30-38 Evens only! Due Monday!

9-3/9-4 Multiplying Binomials Using the infamous FOIL method… also known as DISTRIBUTING!!!

Using the Distributive Property Distribute x + 4 As with the other examples we have seen, we can also use the Distributive Property to find the product of two binomials. Simplify: (2x + 3)(x + 4) 2x2 + 8x + 3x + 12 = 2x2 + 11x + 12 2x2 +8x +3x +12 Now Distribute 2x and 3

Multiplying using FOIL Another way to organize multiplying two binomials is to use FOIL, which stands for, “First, Outer, Inner, Last”. The term FOIL is a memory device for applying the Distributive Property to the product of two binomials. Simplify (3x – 5)(2x + 7) First Outer Inner Last (3x – 5)(2x + 7) = 6x2 + 21x - 10x - 35 = 6x2 + 11x - 35 The product is 6x2 + 11x - 35

Simplify each product. a) (6h – 7)(2h + 3) b) (5m + 2)(8m – 1) c) (9a – 8)(7a + 4) (y – 3)(y + 3) (2x-1)2 12h2 +4h - 21 40m2 + 11m - 2 63a2 -20a - 32 y2 + 3y -3y – 9 or y2 – 9 (2x – 1)(2x – 1) 4x2 - 2x - 2x + 1 4x2 - 4x + 1

Applying Multiplication of Polynomials. area of BIG rectangle = (2x + 5)(3x + 1) area of little rectangle = x(x + 2) area of white region = area of BIG rectangle – area of black rectangle (2x + 5)(3x + 1) – x(x + 2) = 6x2 + 15x + 2x + 5 – x2 – 2x = Combine like terms… 6x2 – x2 + 15x + 2x – 2x + 5 = 5x2 + 15x + 5 Find the area of the white region. Simplify. 2x + 5 x + 2 3x + 1 x Use the Distributive Property to simplify –x(x + 2) Use the FOIL method to simplify (2x + 5)(3x + 1)

Find the area of the shaded region. Simplify. area of BIG rectangle = (5x + 8)(6x + 2) area of little rectangle = 5x(x + 6) area of white region = area of BIG rectangle – area of black rectangle (5x + 8)(6x + 2) - 5x(x + 6)= 30x2 + 10x + 48x + 16 – 5x2 –30x= Combine like terms… Answer: 25x2 + 28x + 16 Find the area of the white region. Simplify. 5x + 8 6x + 2 5x x + 6

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