Upload
others
View
13
Download
0
Embed Size (px)
Citation preview
Holt McDougal Algebra 1
6-5 Multiplying Polynomials6-5 Multiplying Polynomials
Holt Algebra 1
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 Multiplying Polynomials
Multiply polynomials.
Objective
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 Multiplying Polynomials
Multiply.
Example 1C: Multiplying Monomials
−−−−4 53s t
Group factors with like
bases together.
Multiply.
(((( ))))(((( ))))
−−−−22 21
124
ts tt s s
(((( ))))(((( ))))••
•
2 21−12
4ts ts ts
2• •
14 s2 t2 (st) (-12 s t2)
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
When multiplying powers with the same base, keep the base and add the exponents.
x2 •••• x3 = x2+3 = x5
Remember!
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 Multiplying Polynomials
Check It Out! Example 1 Continued
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 McDougal Algebra 1
6-5 Multiplying Polynomials
To multiply a polynomial by a monomial, use the Distributive Property.
Holt McDougal Algebra 1
6-5 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
Multiply.
Distribute 4.
Holt McDougal Algebra 1
6-5 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
Group like bases
together.
Multiply.
Distribute 6pq.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Multiply.
Example 2C: Multiplying a Polynomial by a Monomial
Group like bases
together.
Multiply.
(((( ))))++++21
62
x y xy x y82 2
x y(((( ))))++++22 6
1
2xyyx 8
2
x y x y(((( )))) (((( ))))
++++2 21
62
8xy x y 22
1
2
x2 • x(((( ))))(((( )))) (((( ))))(((( ))))
++++
1• 6
2y • y x2 • x2 y • y2• 8
1
2
3x3y2 + 4x4y3
Distribute .21x y
2
Holt McDougal Algebra 1
6-5 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
Multiply.
Distribute 2.
Holt McDougal Algebra 1
6-5 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
Group like bases
together.
Multiply.
Distribute 3ab.
Holt McDougal Algebra 1
6-5 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
Group like bases
together.
Multiply.
Distribute 5r2s2.
Holt McDougal Algebra 1
6-5 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)
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
Distribute.
Distribute again.
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 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.
Distribute again.
Multiply.
Combine like terms.
Holt McDougal Algebra 1
6-5 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 + 16
x2 – 8x + 16
Write as a product of
two binomials.
Use the FOIL method.
Multiply.
Combine like terms.
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 Multiplying Polynomials
In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5)
Helpful Hint
Holt McDougal Algebra 1
6-5 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.
Distribute again.
Multiply.
Combine like terms.
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 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 + 3××××
6x2 + 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 McDougal Algebra 1
6-5 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.
Distribute x again.
Simplify.
Combine like terms.
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 Multiplying Polynomials
Multiply.
Example 4C: Multiplying Polynomials
(x + 3)3
[x · x + x(3) + 3(x) + (3)(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 McDougal Algebra 1
6-5 Multiplying Polynomials
Example 4C: Multiplying Polynomials 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.
Distribute again.
(x + 3)(x2 + 6x + 9)
Combine like terms.
Holt McDougal Algebra 1
6-5 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 – 11x3 – 4x2 – 21x – 7 Combine like terms.
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 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.
Distribute 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 McDougal Algebra 1
6-5 Multiplying Polynomials
Check It Out! Example 4b
Multiply.
(3x + 2)(x2 – 2x + 5)
(3x + 2)(x2 – 2x + 5)
x2 – 2x + 53x + 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 + 15x
3x3 – 4x2 + 11x + 10Combine like terms by
adding vertically.
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 Multiplying Polynomials
Example 5: Application ContinuedThe 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 McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 Multiplying Polynomials
Check It Out! Example 5 Continued
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 McDougal Algebra 1
6-5 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 McDougal Algebra 1
6-5 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
1
2h2 + 2h