Download Multiplying Polynomials

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
4x2 and -2x3
Using commutative and associative
properties,
4x2(-2x3) = 4(-2)x2x3
= -8x5
Your Turn
1. Multiply
3x5(2x5)
 Solution
3x5(2x5) = 3(2)x5x5
= 6x10
2. Multiply
-2a2b3(5ab2)
 Solution
-2a2b3(5ab2) = -2(5)a2ab3b2
= -10a3b5
Multiplying polynomial by monomial
Multiply: 2x + 4 by 5x
5x(2x + 4) = 5x ∙ 2x + 5x ∙ 4
= 10X2 + 20x
Your Turn
Multiply:
3a2(3a2 – 5a)
Solution:
3a2(3a2 – 5a) = 3a2 ∙ 3a2 – 3a2 ∙ 5a
= 9a4 – 15a3
Multiply:
-2xz2(2x – 3z + 2z2)
Solution:
-2xz2(2x – 3z + 2z2)
= -2xz2 ∙ 2x + (-2xz2) ∙ (-3z) + (-2xz2) ∙ 2z2
= -4x2z2 + 6xz3 + (-4xz4)
= -4x2z2 + 6xz3 – 4xz4
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)
= 6a2 – 12a + 10a – 20
= 6a2 – 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)
= 6a2 + 10a – 12a – 20
= 6a2 – 2a - 20
Example: Squaring a binomial
Multiply: (x + y)2
(x + y)2 = (x + y)(x + y)
= x2 + xy + xy + y2
x2 + 2xy + y2
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)
= x2 - xy - xy + y2
x2 - 2xy + y2
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)
= x2 - xy + xy + y2
x2 - y2
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: (3x2 + 3x – 5)(2x + 3)
Solution:
(2x + 3)(3x2 + 3x – 5)
 = (2x + 3)(3x2 + (2x + 3)3x + (2x + 3)(-5)
= 3x2(2x + 3) + 3x(2x + 3) – 5(2x + 3)
= 6x3 + 9x2 + 6x2 + 9x – 10x – 15
= 6x3 + 15x2 – x - 15
Example
Multiply: (3a2 – 4a + 7)(2a + 5)
Your Turn
Multiply: (3y2 – 5y + 4)(-4y2 – 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)
x2 + 4x + 5x + 20 = x2 + 10x + 9x + 90
x2 + 9x + 20 = x2 + 19x + 90
9x + 20 = 19x + 90
-70 = 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) – x2 = 96
4. Solve the equation.
 (x2 + 8x + 16) – x2= 96
8x = 80
x = 10
5. Check solution.