4.6 Multiplying Polynomials

Objectives  Multiply two or more monomials  Multiply a polynomial and a monomial  Multiply a binomials by a binomial  Multiply a polynomial by a binomial.  Solve an equation that simplifies to a linear equation  Solve an application involving multiplication of polynomials

Multiplying two or more monomials  Multiply 4x 2 and -2x 3  Using commutative and associative properties, 4x 2 (-2x 3 ) = 4(-2)x 2 x 3 = -8x 5

Your Turn 1.Multiply 3x 5 (2x 5 )  Solution 3x 5 (2x 5 ) = 3(2)x 5 x 5 = 6x 10 2.Multiply -2a 2 b 3 (5ab 2 )  Solution -2a 2 b 3 (5ab 2 ) = -2(5)a 2 ab 3 b 2 = -10a 3 b 5

Multiplying polynomial by monomial  Multiply: 2x + 4 by 5x  5x(2x + 4) = 5x ∙ 2x + 5x ∙ 4 = 10X x

Your Turn  Multiply: 3a 2 (3a 2 – 5a)  Solution: 3a 2 (3a 2 – 5a) = 3a 2 ∙ 3a 2 – 3a 2 ∙ 5a = 9a 4 – 15a 3  Multiply: -2xz 2 (2x – 3z + 2z 2 )  Solution: -2xz 2 (2x – 3z + 2z 2 ) = -2xz 2 ∙ 2x + (-2xz 2 ) ∙ (-3z) + (-2xz 2 ) ∙ 2z 2 = -4x 2 z 2 + 6xz 3 + (-4xz 4 ) = -4x2z 2 + 6xz 3 – 4xz 4

Multiplying binomial by binomial  Multiply: (2a – 4)(3a + 5)  Solution: (2a – 4)(3a + 5) = (2a – 4) 3a + (2a – 4) 5 = 3a(2a – 4) + 5(2a – 4) = 3a ∙ 2a + 3a ∙ (-4) + 5 ∙ 2a + 5 ∙ (-4) = 6a 2 – 12a + 10a – 20 = 6a 2 – 2a - 20

Multiplying binomial by binomial  Multiply: (2a – 4)(3a + 5)  Use the First-Outer-Inner-Last (FOIL) method = 2a(3a) + 2a(5) + (-4)(3a) + (-4)(5) = 6a a – 12a – 20 = 6a 2 – 2a - 20

Example: Squaring a binomial  Multiply: (x + y) 2  (x + y) 2 = (x + y)(x + y) = x 2 + xy + xy + y 2 x 2 + 2xy + y 2  The square of the sum of two terms is: square of the first plus twice the product of first and second plus the square of second.

Example: Squaring a binomial  Multiply: (x - y) 2  (x - y) 2 = (x - y)(x - y) = x 2 - xy - xy + y 2 x 2 - 2xy + y 2  The square of the difference of two terms is: square of the first minus twice the product of first and second plus the square of second.

Example: Product of sum and difference  Multiply: (x + y) ( x – y)  (x + y)(x – y) = (x + y)(x - y) = x 2 - xy + xy + y 2 x 2 - y 2  The product of sum and difference of binomials is the square of the first minus the square of the second.

Multiplying polynomial by monomial  Multiply: (3x 2 + 3x – 5)(2x + 3)  Solution: (2x + 3)(3x 2 + 3x – 5)  = (2x + 3)(3x 2 + (2x + 3)3x + (2x + 3)(-5) = 3x 2 (2x + 3) + 3x(2x + 3) – 5(2x + 3) = 6x 3 + 9x 2 + 6x 2 + 9x – 10x – 15 = 6x x 2 – x - 15

Example  Multiply: (3a 2 – 4a + 7)(2a + 5)

Your Turn  Multiply: (3y 2 – 5y + 4)(-4y 2 – 3)  Solution

Solve an equation that simplifies to a linear equation  Solve: (x + 5)(x + 4) = (x + 9)(x + 10)  Solution: (x + 5)(x + 4) = (x + 9)(x + 10) x 2 + 4x + 5x + 20 = x x + 9x + 90 x 2 + 9x + 20 = x x x + 20 = 19x = 10x x = -7

Application involving multiplication of polynomials  A square painting is surrounded by a border 2 inches wide. If the area of the border is 96 square inches, find the dimensions of the painting.

Application involving multiplication of polynomials 1.What am looking for?  dimension of painting: x 2.What is known?  area of border: 96  width of edge: 2 3.Form an equation.  (x + 4)(x + 4) – x 2 = 96 4.Solve the equation.  (x 2 + 8x + 16) – x 2 = 96 8x = 80 x = 10 5.Check solution.