warm up evaluate. 1. 3 2 3. 10 2 simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 ...
TRANSCRIPT
![Page 1: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/1.jpg)
![Page 2: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/2.jpg)
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
![Page 3: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/3.jpg)
Multiply polynomials.
Objective
![Page 4: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/4.jpg)
To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.
![Page 5: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/5.jpg)
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)
![Page 6: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/6.jpg)
Multiply.
Example 1C: Multiplying Monomials
4 53s t
Group factors with like bases together.
Multiply.
22 2112
4ts tt s s
••
•
2 21−12
4t s ts ts
2• •
14 s2 t2 (st) (-12 s t2)
![Page 7: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/7.jpg)
When multiplying powers with the same base, keep the base and add the exponents.
x2 x3 = x2+3 = x5
Remember!
![Page 8: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/8.jpg)
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
![Page 9: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/9.jpg)
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
• • • •
![Page 10: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/10.jpg)
To multiply a polynomial by a monomial, use the Distributive Property.
![Page 11: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/11.jpg)
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.
![Page 12: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/12.jpg)
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.
![Page 13: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/13.jpg)
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
![Page 14: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/14.jpg)
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.
![Page 15: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/15.jpg)
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.
![Page 16: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/16.jpg)
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.
![Page 17: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/17.jpg)
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.
![Page 18: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/18.jpg)
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
![Page 19: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/19.jpg)
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.
![Page 20: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/20.jpg)
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.
![Page 21: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/21.jpg)
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.
![Page 22: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/22.jpg)
In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5)
Helpful Hint
![Page 23: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/23.jpg)
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.
![Page 24: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/24.jpg)
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.
![Page 25: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/25.jpg)
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.
![Page 26: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/26.jpg)
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
![Page 27: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/27.jpg)
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
![Page 28: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/28.jpg)
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.
![Page 29: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/29.jpg)
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.
![Page 30: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/30.jpg)
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.
![Page 31: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/31.jpg)
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.
![Page 32: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/32.jpg)
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.
![Page 33: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/33.jpg)
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.
![Page 34: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/34.jpg)
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
![Page 35: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/35.jpg)
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.
![Page 36: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/36.jpg)
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.
![Page 37: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/37.jpg)
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.
![Page 38: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/38.jpg)
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.
![Page 39: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/39.jpg)
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.
![Page 40: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/40.jpg)
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
![Page 41: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/41.jpg)
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
![Page 42: Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3 2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5 y 4 5656 7. (x 2 ) 4 8. –4(x – 7) –4x + 28 y9y9](https://reader031.vdocuments.site/reader031/viewer/2022032805/56649ee85503460f94bf956f/html5/thumbnails/42.jpg)
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