7-3 Multiply a Polynomial by a Monomial
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Date
Use the Laws of Exponents to multiply monomials algebraically.
Multiply: (5a4b5)(⫺7ab2)
(5a4b5)(⫺7ab2) ⫽ (5)(⫺7)(a4 • a)(b5 • b2)
Multiply the coefficients and
multiply the variables.
⫽ (⫺35a4 1b5 2)
Apply the Law of Exponents for Multiplication.
⫽ ⫺35a5b7
Simplify.
Remember: Laws of Exponents
Product of Powers: am • an ⫽ amn
Power of a Power: (am)n ⫽ am(n)
Simplify: (⫺5x2y6)3
Power of a Product: (ab)m ⫽ ambm
(⫺5x2y6)3 ⫽ (⫺5)3(x2)3(y6)3
Write out the factors of the power.
⫽ (⫺125)x2 • 3y6 • 3
⫽ ⫺125 x6y18
Apply the Law of Exponents for
Power of a Product.
Simplify.
Simplify: 4c2d 5 (⫺6c3d 2 ⫹ 5c4d 2)
4c2d 5 (⫺6c3d 2) ⫹ 4c2d 5 (5c4d 2)
Apply the Distributive Property.
2
3
5
2
2
4(⫺6)(c • c )(d • d ) ⫹ 4(5)(c • c4)(d 5 • d 2)
Apply the Commutative and Associative Properties.
⫺24(c23)(d 5 2) ⫹ 20(c2 4)(d 5 2)
Apply the Law of Exponents for Multiplication.
⫺24c5d 7 ⫹ 20c6d 7
Simplify.
Simplify each expression.
2. (9c6d 4)(3c4d 9)
1. (4a3b7)(8a2b5)
(4)(8)(a3 • a2)(b7 • b5)
32a32b7 5
32a5b12
(
Copyright © by William H. Sadlier, Inc. All rights reserved.
1
)
5. (9f 5g10)3 ⫺ 3 f 5g4
(
(729f 15g30)
7. 6x4y6 ⫺ (12x2y)2(3y4)
(
)
8. 9d 3e8 ⫺ (3de2)2(7de4)
(144x4y2)(3y4)
9d3e8
6x4y6 432x4y6
426x4y6
13. ⫺4a3b3 (⫺3a2b2 ⫺ 2ab3)
12a32b32 8a31b33
12a5b5 8a4b6
)
12. 5x3y4(⫺4x4y ⫹ 7x2y7)
3a2b2(2a3b4) 3a2b2(4ab2)
6(a2 3b2 4) 12(a2 1b2 2)
6a5b6 12a3b4
(4a3b3)(2ab3)
(
15v8w7 (64v9w6)(2vw5)
15v8w7 128v10w11
11. 3a2b2(⫺2a3b4 + 4ab2)
23g6h4 (81g8h12)(4gh6)
23g6h4 324g9h18
)
1 t14u28
(512t6u18)
16
32t14 6u28 18
32t 20u46
9. ⫺15v8w7 ⫺ (4v3w2)3(2vw5)
(9d2e4)(7de4)
9d3e8 – 63d3e8
54d3e8
10. ⫺23g6h4 ⫺ (3g2h3)4(4gh6)
1
6. ⫺ 4 t7u14 2(8t 2u6)3
2
1 f 10g8
9
81f 1510g308
81f 25g38
(12)(5)(m9 • m6)(n8 • n12)
60m9 6n8 12
60m15n20
4a3b3(3a2b2)
(2)(7)(x8 • x10)(y5 • y2)
14x8 10y5 2
14x18y7
(9)(3)(c6 • c4)(d4 • d9)
27c64d4 9
27c10d13
4. (12m9n8)(⫺5m6n12)
6x4y6
3. (⫺2x8y5)(7x10y2)
5x3y4(4x4y) 5x3y4(7x2y7)
20(x3 4y4 1) 35(x3 2y47)
20x7y5 35x5y11
14. 7m2n3 (⫺5m2n3 ⫺ 9m3n4)
Lesson 7-3, pages 182–183.
7m2n3(5m2n3) 7m2n3(9m3n4)
35m2 2n3 3 63m2 3n34
35m4n6 63m5n7
Chapter 7
173
For More Practice Go To:
Simplify.
15. 5d 2f 4(3d 5f 2 ⫺ 5d 2f 3 ⫹ 7df 5)
16. 3x3y5(9x6y4 ⫺ 4x2y4 ⫹ 3xy6)
5d 2 f 4 (3d 5f 2) 5d 2 f 4 (5d 2 f 3) 5d 2 f 4 (7df 5)
15(d 2 5)( f 4 2) (25)(d 2 2)(f 4 3) 35(d 2 1)(f 4 5)
15d 7f 6 25d 4f 7 35d 3f 9
17. 2.4a7b2(1.3a5b ⫹ 2.5a3b2 ⫹ 0.4a2)
18. 3.2c8d4(0.9c6d ⫹ 1.5c4d3 ⫹ 0.2d 5)
2.4a7b2(1.3a5b) 2.4a7b2(2.5a3b2) 2.4a7b2(0.4a2)
3.12(a7 5)(b2 1) 6(a7 3)(b22) 0.96(a7 2)(b2)
3.12a12b3 6a10b4 0.96a9b2
1
(
1
) (
20. 3 rs2 15r2s2 ⫹ 12rs ⫹ 4 s) ⫹ 8r2s3 ⫺ 2 rs3
)
1
1 2
1
1
rs (15r 2s2) rs2(12rs) rs2 s
3
3
3
4
1 3
1
3
4
2
3
2
3
5r s 4r s rs 8r s rs3
12
2
5 3
3
4
2
3
5r s 12r s rs
12
1
1 2
1
1
x y(20xy) x2y(8x) x2y y
2
2
2
2
1 2 2
3
3
2
3
10x y 4x y x y 3x3y2 x2y2
4
4
1 2 2
3
2
3
13x y 4x y x y
2
( )
) (
3.2c8d4(0.9c6d) 3.2c8d4(1.5c4d3) 3.2c8d4(0.2d 5)
2.88(c8 6)(d4 1) 4.8(c8 4)(d4 3) 0.64(c8)(d4 5)
2.88c14d5 4.8c12d 7 0.64c8d 9
)
3
19. 2 x2y 20xy ⫹ 8x ⫹ 2 y ⫹ 3x3y2 ⫺ 4 x2y2
(
3x3y5(9x6y4) 3x3y5(4x2y4) 3x3y5(3xy6)
27(x3 6)(y5 4) (12)(x3 2)(y5 4) 9(x3 1)(y5 6)
27x9y9 12x5y9 9x4y11
1
(
1
(
1
)
( )
) (
(
)
Solve. Show your work.
21. Write a polynomial in simplest form
to express the shaded area.
x
3x
5x ⫹ 2y
7x
(5x 2y)(3x) (3x 3y)(x)
15x2 6xy (3x2 3xy)
12x2 9xy
23. Simplify.
(x3a)b(x7ab) ⫹ (xa)5a(x2a)
2
(x3ab )(x7ab ) (x5a )(x2a)
2
(x3ab 7ab) (x5a 2a)
2 2a
10ab
5a
x
x
Chapter 7
(9x2)2 (7x)2; (92)(x2• 2) (72)(x1• 2)
81x4 49x2
Copyright © by William H. Sadlier, Inc. All rights reserved.
9x 2
3x ⫺ 3y
174
22. Write a polynomial in simplest form
to express the shaded area in terms of ␲.