holt algebra 1 7-7 multiplying polynomials 7-7 multiplying polynomials holt algebra 1 warm up warm...
TRANSCRIPT
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
Warm UpEvaluate.
1. 32
3. 102
Simplify.
4. 23 24
6. (53)2
9 16
100
27
2. 24
5. y5 y4
56 7. (x2)4
8. –4(x – 7) –4x + 28
y9
x8
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
Check It Out! Example 1
Multiply.
a. (3x3)(6x2)
(3x3)(6x2)
(3 6)(x3 x2)
18x5
Group factors with like bases together.
Multiply.
Group factors with like bases together.
Multiply.
b. (2r2t)(5t3)
(2r2t)(5t3)
(2 5)(r2)(t3 t)
10r2t4
Holt Algebra 1
7-7 Multiplying Polynomials
Check It Out! Example 1
Multiply.
Group factors with like bases together.
Multiply.
c.
4 52 2112
3x zy zx y
3
2112
3x y x z y z
2 4 53
3 22 4 5112 z
3zx x y y
7554x y z
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
Multiply.
Example 2C: Multiplying a Polynomial by a Monomial
Group like bases together.
Multiply.
216
2x y xy x y8 2 2
x y 22 612
xyyx 8 2
x y x y
2 216
28xy x y 22
12
x2 • x
1
• 62
y • y x2 • x2 y • y2• 8
12
3x3y2 + 4x4y3
Distribute .21x y
2
Holt Algebra 1
7-7 Multiplying Polynomials
Check It Out! Example 2
Multiply.
a. 2(4x2 + x + 3)
2(4x2 + x + 3)
2(4x2) + 2(x) + 2(3)
8x2 + 2x + 6
Distribute 2.
Multiply.
Holt Algebra 1
7-7 Multiplying Polynomials
Check It Out! Example 2
Multiply.
b. 3ab(5a2 + b)
3ab(5a2 + b)
(3ab)(5a2) + (3ab)(b)
(3 5)(a a2)(b) + (3)(a)(b b)
15a3b + 3ab2
Distribute 3ab.
Group like bases together.
Multiply.
Holt Algebra 1
7-7 Multiplying Polynomials
Check It Out! Example 2
Multiply.
c. 5r2s2(r – 3s)
5r2s2(r – 3s)
(5r2s2)(r) – (5r2s2)(3s)
(5)(r2 r)(s2) – (5 3)(r2)(s2 s)
5r3s2 – 15r2s3
Distribute 5r2s2.
Group like bases together.
Multiply.
Holt Algebra 1
7-7 Multiplying Polynomials
To multiply a binomial by a binomial, you can apply the Distributive Property more than once:
(x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3.
Distribute x and 3 again.
Multiply.
Combine like terms.
= x(x + 2) + 3(x + 2)
= x(x) + x(2) + 3(x) + 3(2)
= x2 + 2x + 3x + 6
= x2 + 5x + 6
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
Check It Out! Example 3a
Multiply.
(a + 3)(a – 4)
(a + 3)(a – 4)
a(a – 4)+3(a – 4)
a(a) + a(–4) + 3(a) + 3(–4)
a2 – a – 12
a2 – 4a + 3a – 12
Distribute a and 3.
Distribute a and 3 again.
Multiply.
Combine like terms.
Holt Algebra 1
7-7 Multiplying Polynomials
Check It Out! Example 3b
Multiply.
(x – 3)2
(x – 3)(x – 3)
(x x) + (x(–3)) + (–3 x)+ (–3)(–3) ●
x2 – 3x – 3x + 9
x2 – 6x + 9
Write as a product of two binomials.
Use the FOIL method.
Multiply.
Combine like terms.
Holt Algebra 1
7-7 Multiplying Polynomials
Check It Out! Example 3c
Multiply.
(2a – b2)(a + 4b2)
(2a – b2)(a + 4b2)
2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2)
2a2 + 8ab2 – ab2 – 4b4
2a2 + 7ab2 – 4b4
Use the FOIL method.
Multiply.
Combine like terms.
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
You can also use a rectangle model to multiply polynomials with more than two terms. This is similar to finding the area of a rectangle with length (2x2 + 10x – 6) and width (5x + 3):
2x2 +10x –6
10x3 50x2 –30x
30x6x2 –18
5x
+3
Write the product of the monomials in each row and column:
To find the product, add all of the terms inside the rectangle by combining like terms and simplifying if necessary. 10x3 + 6x2 + 50x2 + 30x – 30x – 18
10x3 + 56x2 – 18
Holt Algebra 1
7-7 Multiplying Polynomials
Another method that can be used to multiply polynomials with more than two terms is the vertical method. This is similar to methods used to multiply whole numbers.
2x2 + 10x – 6
5x + 36x2 + 30x – 18
+ 10x3 + 50x2 – 30x 10x3 + 56x2 + 0x – 18
10x3 + 56x2 + – 18
Multiply each term in the top polynomial by 3.
Multiply each term in the top polynomial by 5x, and align like terms.
Combine like terms by adding vertically.
Simplify.
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
(2x – 5)(–4x2 – 10x + 3)
(2x – 5)(–4x2 – 10x + 3)
–4x2 – 10x + 32x – 5x
20x2 + 50x – 15+ –8x3 – 20x2 + 6x
–8x3 + 56x – 15
Multiply each term in the top polynomial by –5.
Multiply each term in the top polynomial by 2x, and align like terms.
Combine like terms by adding vertically.
Holt Algebra 1
7-7 Multiplying Polynomials
Multiply.
Example 4C: 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 4C: Multiplying Polynomials
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
Example 4D: Multiplying Polynomials
Multiply.(3x + 1)(x3 – 4x2 – 7)
x3 4x2 –7
3x4 12x3 –21x
4x2x3 –7
3x
+1
3x4 + 12x3 + x3 + 4x2 – 21x – 7
Write the product of the monomials in each row and column.
Add all terms inside the rectangle.
3x4 + 13x3 + 4x2 – 21x – 7 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 4a
Multiply.
(x + 3)(x2 – 4x + 6)
(x + 3 )(x2 – 4x + 6)
x(x2 – 4x + 6) + 3(x2 – 4x + 6)
Distribute x and 3.
Distribute x and 3 again.
x(x2) + x(–4x) + x(6) +3(x2) +3(–4x) +3(6)
x3 – 4x2 + 3x2 +6x – 12x + 18
x3 – x2 – 6x + 18
Simplify.
Combine like terms.
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.
Holt Algebra 1
7-7 Multiplying Polynomials
Check It Out! Example 5
The length of a rectangle is 4 meters shorter than its width.
a. Write a polynomial that represents the area of the rectangle.
Write the formula for the area of a rectangle.
Substitute x – 4 for l and x for w.
A = l w
A = l w
A = x(x – 4)
Multiply.A = x2 – 4x
The area is represented by x2 – 4x.
Holt Algebra 1
7-7 Multiplying Polynomials
Check It Out! Example 5
The length of a rectangle is 4 meters shorter than its width.
b. Find the area of a rectangle when the width is 6 meters. A = x2 – 4x
A = x2 – 4x
A = 36 – 24
A = 12
Write the formula for the area of a rectangle whose length is 4 meters shorter than width .
Substitute 6 for x.
Simplify.
Combine terms.
The area is 12 square meters.
A = 62 – 4 6
Holt Algebra 1
7-7 Multiplying Polynomials
Lesson Quiz: Part I
Multiply.
1. (6s2t2)(3st)
2. 4xy2(x + y)
3. (x + 2)(x – 8)
4. (2x – 7)(x2 + 3x – 4)
5. 6mn(m2 + 10mn – 2)
6. (2x – 5y)(3x + y)
4x2y2 + 4xy3
18s3t3
x2 – 6x – 16
2x3 – x2 – 29x + 28
6m3n + 60m2n2 – 12mn
6x2 – 13xy – 5y2
Holt Algebra 1
7-7 Multiplying Polynomials
Lesson Quiz: Part II
7. A triangle has a base that is 4cm longer than its height.a. Write a polynomial that represents the area
of the triangle.
b. Find the area when the height is 8 cm.
48 cm2
12
h2 + 2h