Multiplying a Polynomial by a Monomial, Multiplying Polynomials (7-6, 7-7) Objective: Multiply a polynomial by a monomial. Solve equations involving the products of monomials and polynomials. Multiply polynomials by using the Distributive Property. Multiply polynomials using the FOIL method.

Polynomial Multiplied by Monomial To find the product of a polynomial and a monomial, you can use the Distributive Property.

Example 1 Find 6y(4y 2 – 9y – 7) =6y(4y 2 ) + 6y(-9y) + 6y(-7) =24y 3 – 54y 2 – 42y

Check Your Progress Choose the best answer for the following. – Find 3x(2x 2 + 3x + 5). A.6x 2 + 9x + 15 B.6x 3 + 9x x C.5x 3 + 6x 2 + 8x D.6x 2 + 3x + 5 3x(2x 2 ) + 3x(3x) + 3x(5)

Polynomial Multiplied by Monomial We can use this same method more than once to simplify large expressions.

Example 2 Simplify 3(2t 2 – 4t – 15) + 6t(5t + 2). =3(2t 2 ) + 3(-4t) + 3(-15) + 6t(5t) + 6t(2) =6t 2 – 12t – t t =36t 2 – 45

Check Your Progress Choose the best answer for the following. – Simplify 5(4y 2 + 5y – 2) + 2y(4y + 3). A.4y 2 + 9y + 1 B.8y 2 + 5y – 6 C.20y 2 + 9y + 6 D.28y y – 10 5(4y 2 ) + 5(5y) + 5(-2) + 2y(4y) + 2y(3) 20y y – y 2 + 6y

Polynomial Multiplied by Monomial We can use the Distributive Property to multiply monomials by polynomials and solve real world problems.

Example 3 Admission to the Super Fun Amusement Park is $10. Once in the park, super rides are an additional $3 each, and regular rides are an additional $2. Wyome goes to the park and rides 15 rides, of which s of those 15 are super rides. Find the cost in dollars if Wyome rode 9 super rides. – C = 3s + 2(15 – s) + 10 – C = 3s + 30 – 2s + 10 – C = s + 40 – C = – C = $49

Check Your Progress Choose the best answer for the following. – The Fosters own a vacation home that they rent throughout the year. The rental rate during peak season is $120 per day and the rate during the off-peak season is $70 per day. Last year they rented the house 210 days, p of which were during peak season. Determine how much rent the Fosters received if p is equal to 130. A.$120,000 B.$21,200 C.$70,000 D.$210,000 R = 120p + 70(210 – p) R = 120p + 14,700 – 70p R = 50p + 14,700 R = 50(130) + 14,700

Solve Equations with Polynomial Expressions We can use the Distributive Property to solve equations that involve the products of monomials and polynomials.

Example 4 Solve b(12 + b) – 7 = 2b + b(-4 + b). 12b + b 2 – 7 = 2b – 4b + b 2 12b + b 2 – 7 = -2b + b 2 -b 2 -b 2 12b – 7 = -2b +2b +2b 14b – 7 = b = b = ½

Check Your Progress Choose the best answer for the following. – Solve x(x + 2) + 2x(x – 3) + 7 = 3x(x – 5) – 12. A.- 19 / 11 B.-2 C. 21 / 11 D.0 x 2 + 2x + 2x 2 – 6x + 7 = 3x 2 – 15x – 12 3x 2 – 4x + 7 = 3x 2 – 15x – 12 -3x 2 -4x + 7 = -15x – x 11x + 7 = x = -19

Multiply Binomials To multiply two binomials such as h – 32 and ½ h + 11, the Distributive Property is used. Binomials can be multiplied horizontally or by the box method.

Example 5 Find each product. a.(y + 8)(y – 4) =y(y – 4) + 8(y – 4) =y 2 – 4y + 8y – 32 =y 2 + 4y – 32

Example 5 Find each product. b.(2x + 1)(x + 6) 2x +1 x+6 2x 2 +12x +x+6 2x x + 6

Check Your Progress Choose the best answer for the following. A.Find (c + 2)(c – 4). A.c 2 – 6c + 8 B.c 2 – 4c – 8 C.c 2 – 2c + 8 D.c 2 – 2c – 8 c(c – 4) + 2(c – 4) c 2 – 4c + 2c – 8

Check Your Progress Choose the best answer for the following. B.Find (x + 3)(4x – 1). A.4x 2 – 11x – 3 B.4x x – 3 C.4x x – 3 D.4x x – 3 x +3 4x 4x 2 -x +12x-3

FOIL Method A shortcut version of the Distributive Property for multiplying binomials is called the FOIL method. To multiply two binomials, find the sum of the products of F the first terms, O the outer terms, I the inner terms, and L the last terms. (x + 4)(x – 2) F 0 I L = x(x) + x(-2) + 4(x) + 4(-2) = x 2 – 2x + 4x – 8 = x 2 + 2x – 8

Example 6 Find each product. a.(z – 6)(z – 12) F 0 I L = z(z) + z(-12) – 6(z) – 6(-12) = z 2 – 12z – 6z + 72 = z 2 – 18z + 72

Example 6 Find each product. b.(5x – 4)(2x + 8) F 0 I L = 5x(2x) + 5x(8) – 4(2x) – 4(8) = 10x x – 8x – 32 = 10x x – 32

Check Your Progress Choose the best answer for the following. A.Find (x + 2)(x – 3). A.x 2 + x – 6 B.x 2 – x – 6 C.x 2 + x + 6 D.x 2 + x + 5 = x(x) + x(-3) + 2(x) + 2(-3) = x 2 – 3x + 2x – 6

Check Your Progress Choose the best answer for the following. B.Find (3x + 5)(2x – 6). A.5x 2 – 8x + 30 B.6x x – 1 C.6x 2 – 8x – 30 D.6x – 30 = 3x(2x) + 3x(-6) + 5(2x) + 5(-6) = 6x 2 – 18x + 10x – 30

Real-World Examples Notice that when two linear expressions are multiplied, the result is a quadratic expression. A quadratic expression is an expression in one variable with a degree of 2. When three linear expressions are multiplied, the result has a degree of 3. The FOIL method can be used to find an expression that represents the area of an object when the lengths of the sides are given as binomials.

Example 7 A patio in the shape of the triangle shown below is being built in Lavali’s backyard. The dimensions given are in feet. The area A of the triangle is one half the height h times the base b. Write an expression for the area of the patio. – A = ½ bh – A = ½ (6x + 4)(x – 7) – A = ½ (6x 2 – 42x + 4x – 28) – A = ½ (6x 2 – 38x – 28) – A = 3x 2 – 19x – 14 ft 2

Check Your Progress Choose the best answer for the following. – The area of a rectangle is the measure of the base times the height. Write an expression for the area of the rectangle. A.7x + 3 units 2 B.12x x + 2 units 2 C.12x 2 + 8x + 2 units 2 D.7x x + 3 units 2 A = lw = (4x + 1)(3x + 2) = 12x 2 + 8x + 3x + 2