Embed Size (px)
Citation preview
Holt Algebra 1
7-7 Multiplying Polynomials7-7 Multiplying Polynomials
Holt Algebra 1
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 1
7-7 Multiplying Polynomials
Multiply polynomials.
Objective
Holt Algebra 1
7-7 Multiplying Polynomials
To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.
Holt Algebra 1
7-7 Multiplying Polynomials
Multiply.
Example 1: Multiplying Monomials
A. (6y3)(3y5)
(6y3)(3y5)
18y8
Group factors with like bases together.
B. (3mn2) (9m2n)
(3mn2)(9m2n)
27m3n3
Multiply.
Group factors with like bases together.
Multiply.
(6 3)(y3 y5)
(3 9)(m m2)(n2 n)
Holt Algebra 1
7-7 Multiplying Polynomials
Multiply.
Example 1C: Multiplying Monomials
2 2 2112
4s t st st
4 53s t
Group factors with like bases together.
Multiply.
22 2112
4ts tt s s
2 2112
4t s ts ts
2
Holt Algebra 1
7-7 Multiplying Polynomials
When multiplying powers with the same base, keep the base and add the exponents.
x2 x3 = x2+3 = x5
Remember!
Holt Algebra 1
7-7 Multiplying Polynomials
To multiply a polynomial by a monomial, use the Distributive Property.
Holt Algebra 1
7-7 Multiplying Polynomials
Multiply.
Example 2A: Multiplying a Polynomial by a Monomial
4(3x2 + 4x – 8)
4(3x2 + 4x – 8)
(4)3x2 +(4)4x – (4)8
12x2 + 16x – 32
Distribute 4.
Multiply.
Holt Algebra 1
7-7 Multiplying Polynomials
6pq(2p – q)
(6pq)(2p – q)
Multiply.
Example 2B: Multiplying a Polynomial by a Monomial
(6pq)2p + (6pq)(–q)
(6 2)(p p)(q) + (–1)(6)(p)(q q)
12p2q – 6pq2
Distribute 6pq.
Group like bases together.
Multiply.
Holt Algebra 1
7-7 Multiplying Polynomials
Another method for multiplying binomials is called the FOIL method.
4. Multiply the Last terms. (x + 3)(x + 2) 3 2 = 6
3. Multiply the Inner terms. (x + 3)(x + 2) 3 x = 3x
2. Multiply the Outer terms. (x + 3)(x + 2) x 2 = 2x
F
O
I
L
(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6
F O I L
1. Multiply the First terms. (x + 3)(x + 2) x x = x2
Holt Algebra 1
7-7 Multiplying Polynomials
Multiply.
Example 3A: Multiplying Binomials
(s + 4)(s – 2)
(s + 4)(s – 2)
s(s – 2) + 4(s – 2)
s(s) + s(–2) + 4(s) + 4(–2)
s2 – 2s + 4s – 8
s2 + 2s – 8
Distribute s and 4.
Distribute s and 4 again.
Multiply.
Combine like terms.
Holt Algebra 1
7-7 Multiplying Polynomials
Multiply.
Example 3B: Multiplying Binomials
(x – 4)2
(x – 4)(x – 4)
(x x) + (x (–4)) + (–4 x) + (–4 (–4))
x2 – 4x – 4x + 8
x2 – 8x + 8
Write as a product of two binomials.
Use the FOIL method.
Multiply.
Combine like terms.
Holt Algebra 1
7-7 Multiplying Polynomials
Example 3C: Multiplying Binomials
Multiply.
(8m2 – n)(m2 – 3n)
8m2(m2) + 8m2(–3n) – n(m2) – n(–3n)
8m4 – 24m2n – m2n + 3n2
8m4 – 25m2n + 3n2
Use the FOIL method.
Multiply.
Combine like terms.
Holt Algebra 1
7-7 Multiplying Polynomials
In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5)
Helpful Hint
Holt Algebra 1
7-7 Multiplying Polynomials
To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x2 + 10x – 6):
(5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6)
= 10x3 + 50x2 – 30x + 6x2 + 30x – 18
= 10x3 + 56x2 – 18
Holt Algebra 1
7-7 Multiplying Polynomials
Multiply.
Example 4A: Multiplying Polynomials
(x – 5)(x2 + 4x – 6)
(x – 5 )(x2 + 4x – 6)
x(x2 + 4x – 6) – 5(x2 + 4x – 6)
x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6)
x3 + 4x2 – 5x2 – 6x – 20x + 30
x3 – x2 – 26x + 30
Distribute x and –5.
Distribute x and −5 again.
Simplify.
Combine like terms.
Holt Algebra 1
7-7 Multiplying Polynomials
Multiply.
Example 4B: Multiplying Polynomials
(x + 3)3
[(x + 3)(x + 3)](x + 3)
[x(x+3) + 3(x+3)](x + 3)
(x2 + 3x + 3x + 9)(x + 3)
(x2 + 6x + 9)(x + 3)
Write as the product of three binomials.
Use the FOIL method on the first two factors.
Multiply.
Combine like terms.
Holt Algebra 1
7-7 Multiplying Polynomials
Example 4B: Continued
Multiply.
(x + 3)3
x3 + 6x2 + 9x + 3x2 + 18x + 27
x3 + 9x2 + 27x + 27
x(x2) + x(6x) + x(9) + 3(x2) + 3(6x) + 3(9)
x(x2 + 6x + 9) + 3(x2 + 6x + 9)
Use the Commutative Property of Multiplication.
Distribute the x and 3.
Distribute the x and 3 again.
(x + 3)(x2 + 6x + 9)
Combine like terms.
Holt Algebra 1
7-7 Multiplying Polynomials
A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3, or 6 terms before simplifying.
Helpful Hint
Holt Algebra 1
7-7 Multiplying Polynomials
Check It Out! Example 4b
Multiply.
(3x + 2)(x2 – 2x + 5)
(3x + 2)(x2 – 2x + 5)
x2 – 2x + 5 3x + 2
Multiply each term in the top polynomial by 2.
Multiply each term in the top polynomial by 3x, and align like terms.2x2 – 4x + 10
+ 3x3 – 6x2 + 15x3x3 – 4x2 + 11x + 10
Combine like terms by adding vertically.
Holt Algebra 1
7-7 Multiplying Polynomials
Example 5: ApplicationThe width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.
a. Write a polynomial that represents the area of the base of the prism.
Write the formula for the area of a rectangle.
Substitute h – 3 for w and h + 4 for l.
A = l w
A = l w
A = (h + 4)(h – 3)
Multiply.A = h2 + 4h – 3h – 12
Combine like terms.A = h2 + h – 12
The area is represented by h2 + h – 12.
Holt Algebra 1
7-7 Multiplying Polynomials
Example 5: ApplicationThe width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.
b. Find the area of the base when the height is 5 ft.
A = h2 + h – 12
A = h2 + h – 12
A = 52 + 5 – 12
A = 25 + 5 – 12
A = 18
Write the formula for the area the base of the prism.
Substitute 5 for h.
Simplify.
Combine terms.
The area is 18 square feet.
