Multiplying Polynomials

5.6. MULTIPLYING POLYNOMIALS
5.6
309
Multiplying Polynomials
1. Use the commutative and associative properties to change the order and
regroup.
−3(7r) = [(−3)(7)]r
= −21r
Reorder. Regroup.
Multiply: (−3)(7) = −21
3. Use the commutative and associative properties to change the order and
regroup.
(−9b3 )(−8b6 ) = [(−9)(−8)](b3 b6 )
= 72b
9
Reorder. Regroup.
Multiply: (−9)(−8) = 72, b3 b6 = b9 .
5. Use the commutative and associative properties to change the order and
regroup.
(−7r2 t4 )(7r5 t2 )
= [(−7)(7)](r2 r5 )(t4 t2 )
7 6
Reorder. Regroup.
Multiply: (−7)(7) = −49, r2 r5 = r7 ,
= −49r t
and t4 t2 = t6
7. Use the commutative and associative properties to change the order and
regroup.
(−5b2 c9 )(−8b4 c4 )
= [(−5)(−8)](b2 b4 )(c9 c4 )
6 13
Reorder. Regroup.
Multiply: (−5)(−8) = 40, b2 b4 = b6 ,
= 40b c
and c9 c4 = c13
9. Use the commutative and associative properties to change the order and
regroup.
(−8v 3 )(4v 4 ) = [(−8)(4)](v 3 v 4 )
= −32v
7
Reorder. Regroup.
Multiply: (−8)(4) = −32, v 3 v 4 = v 7 .
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11. We need to first distribute the 9 times each term of the polynomial. Then
we multiply the resulting monomials mentally.
9(−2b2 + 2b + 9) = 9(−2b2 ) + 9(2b) + 9(9)
= −18b2 + 18b + 81
Alternate solution. Note that it is more efficient to distribute 9 times each
term of −2b2 + 2b + 9, performing the calculations mentally as you go.
9(−2b2 + 2b + 9) = −18b2 + 18b + 81
13. We need to first distribute the −4 times each term of the polynomial.
Then we multiply the resulting monomials mentally.
−4(10t2 − 7t − 6) = −4(10t2 ) − (−4)(7t) − (−4)(6)
= −40t2 − (−28t) − (−24)
= −40t2 + 28t + 24
Alternate solution. Note that it is more efficient to distribute −4 times each
term of 10t2 − 7t − 6, performing the calculations mentally as you go.
−4(10t2 − 7t − 6) = −40t2 + 28t + 24
15. We need to first distribute the −8u2 times each term of the polynomial.
Then we multiply the resulting monomials mentally.
− 8u2 (−7u3 − 8u2 − 2u + 10)
= −8u2 (−7u3 ) − (−8u2 )(8u2 ) − (−8u2 )(2u) + (−8u2 )(10)
= 56u5 − (−64u4 ) − (−16u3) + (−80u2 )
= 56u5 + 64u4 + 16u3 − 80u2
Alternate solution. Note that it is more efficient to distribute −8u2 times
each term of −7u3 − 8u2 − 2u + 10, performing the calculations mentally as
you go.
−8u2 (−7u3 − 8u2 − 2u + 10) = 56u5 + 64u4 + 16u3 − 80u2
17. We need to first distribute the 10s2 times each term of the polynomial.
Then we multiply the resulting monomials mentally.
10s2 (−10s3 + 2s2 + 2s + 8) = 10s2 (−10s3 ) + 10s2 (2s2 ) + 10s2 (2s) + 10s2 (8)
= −100s5 + 20s4 + 20s3 + 80s2
Alternate solution. Note that it is more efficient to distribute 10s2 times
each term of −10s3 + 2s2 + 2s + 8, performing the calculations mentally as you
go.
10s2 (−10s3 + 2s2 + 2s + 8) = −100s5 + 20s4 + 20s3 + 80s2
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19. We need to first distribute the 2st times each term of the polynomial.
Then we multiply the resulting monomials mentally.
2st(−4s2 + 8st − 10t2 ) = 2st(−4s2 ) + 2st(8st) − 2st(10t2 )
= −8s3 t + 16s2 t2 − 20st3
Alternate solution. Note that it is more efficient to distribute 2 times each
term of −4s2 + 8st − 10t2 , performing the calculations mentally as you go.
2st(−4s2 + 8st − 10t2 ) = −8s3 t + 16s2 t2 − 20st3
21. We need to first distribute the −2uw times each term of the polynomial.
Then we multiply the resulting monomials mentally.
−2uw(10u2 − 7uw − 2w2 ) = −2uw(10u2) − (−2uw)(7uw) − (−2uw)(2w2 )
= −20u3w − (−14u2 w2 ) − (−4uw3 )
= −20u3w + 14u2 w2 + 4uw3
Alternate solution. Note that it is more efficient to distribute −2 times each
term of 10u2 − 7uw − 2w2 , performing the calculations mentally as you go.
−2uw(10u2 − 7uw − 2w2 ) = −20u3w + 14u2w2 + 4uw3
23. Let’s imagine that (−9x − 4)(−3x + 2) has the form (b + c)a and multiply
−3x + 2 times both terms of −9x − 4.
(−9x − 4)(−3x + 2) = −9x(−3x + 2) − 4(−3x + 2)
Now we distribute monomials times polynomials, then combine like terms.
= 27x2 − 18x + 12x − 8
= 27x2 − 6x − 8
Thus, (−9x − 4)(−3x + 2) = 27x2 − 6x − 8
25. Let’s imagine that (3x + 8)(3x − 2) has the form (b + c)a and multiply
3x − 2 times both terms of 3x + 8.
(3x + 8)(3x − 2) = 3x(3x − 2) + 8(3x − 2)
Now we distribute monomials times polynomials, then combine like terms.
= 9x2 − 6x + 24x − 16
= 9x2 + 18x − 16
Thus, (3x + 8)(3x − 2) = 9x2 + 18x − 16
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27. Let’s imagine that (2x − 1)(−6x2 + 4x + 5) has the form (b + c)a and
multiply −6x2 + 4x + 5 times both terms of 2x − 1.
(2x − 1)(−6x2 + 4x + 5) = 2x(−6x2 + 4x + 5) − 1(−6x2 + 4x + 5)
Now we distribute monomials times polynomials, then combine like terms.
= −12x3 + 8x2 + 10x + 6x2 − 4x − 5
= −12x3 + 14x2 + 6x − 5
Thus, (2x − 1)(−6x2 + 4x + 5) = −12x3 + 14x2 + 6x − 5
29. Let’s imagine that (x−6)(−2x2 −4x−4) has the form (b+c)a and multiply
−2x2 − 4x − 4 times both terms of x − 6.
(x − 6)(−2x2 − 4x − 4) = x(−2x2 − 4x − 4) − 6(−2x2 − 4x − 4)
Now we distribute monomials times polynomials, then combine like terms.
= −2x3 − 4x2 − 4x + 12x2 + 24x + 24
= −2x3 + 8x2 + 20x + 24
Thus, (x − 6)(−2x2 − 4x − 4) = −2x3 + 8x2 + 20x + 24
31. First, multiply 8u times each term of 8u − 9w, then multiply −9w times
each term of 8u − 9w. Finally, combine like terms.
(8u − 9w)(8u − 9w) = 8u(8u − 9w) − 9w(8u − 9w)
= 64u2 − 72uw − 72uw + 81w2
= 64u2 − 144uw + 81w2
Thus, (8u − 9w)(8u − 9w) = 64u2 − 144uw + 81w2 .
33. First, multiply 9r times each term of 3r − 9t, then multiply −7t times each
term of 3r − 9t. Finally, combine like terms.
(9r − 7t)(3r − 9t) = 9r(3r − 9t) − 7t(3r − 9t)
= 27r2 − 81rt − 21rt + 63t2
= 27r2 − 102rt + 63t2
Thus, (9r − 7t)(3r − 9t) = 27r2 − 102rt + 63t2 .
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35. First, multiply 4r times each term of −10r2 + 10rs − 7s2 , then multiply
−10s times each term of −10r2 + 10rs − 7s2 . Finally, combine like terms.
(4r − 10s)(−10r2 + 10rs − 7s2 ) = 4r(−10r2 + 10rs − 7s2 ) − 10s(−10r2 + 10rs − 7s2 )
= −40r3 + 40r2 s − 28rs2 + 100r2 s − 100rs2 + 70s3
= −40r3 + 140r2 s − 128rs2 + 70s3
Thus, (4r − 10s)(−10r2 + 10rs − 7s2 ) = −40r3 + 140r2 s − 128rs2 + 70s3 .
37. First, multiply 9x times each term of 4x2 − 4xz − 10z 2, then multiply −2z
times each term of 4x2 − 4xz − 10z 2 . Finally, combine like terms.
(9x − 2z)(4x2 − 4xz − 10z 2 ) = 9x(4x2 − 4xz − 10z 2 ) − 2z(4x2 − 4xz − 10z 2)
= 36x3 − 36x2 z − 90xz 2 − 8x2 z + 8xz 2 + 20z 3
= 36x3 − 44x2 z − 82xz 2 + 20z 3
Thus, (9x − 2z)(4x2 − 4xz − 10z 2 ) = 36x3 − 44x2 z − 82xz 2 + 20z 3.
39. First, write 9r + 3t as a factor two times.
(9r + 3t)2 = (9r + 3t)(9r + 3t)
Next, multiply 9r times each term of 9r + 3t, then multiply 3t times each term
of 9r + 3t. Finally, combine like terms.
= 9r(9r + 3t) + 3t(9r + 3t)
= 81r2 + 27rt + 27rt + 9t2
= 81r2 + 54rt + 9t2
Thus, (9r + 3t)2 = 81r2 + 54rt + 9t2 .
41. First, multiply 4y times each term of 4y − 5z, then multiply 5z times each
term of 4y − 5z. Finally, combine like terms.
(4y + 5z)(4y − 5z) = 4y(4y − 5z) + 5z(4y − 5z)
= 16y 2 − 20yz + 20yz − 25z 2
= 16y 2 − 25z 2
Thus, (4y + 5z)(4y − 5z) = 16y 2 − 25z 2.
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43. First, multiply 7u times each term of 7u − 8v, then multiply 8v times each
term of 7u − 8v. Finally, combine like terms.
(7u + 8v)(7u − 8v) = 7u(7u − 8v) + 8v(7u − 8v)
= 49u2 − 56uv + 56uv − 64v 2
= 49u2 − 64v 2
Thus, (7u + 8v)(7u − 8v) = 49u2 − 64v 2 .
45. First, write 7b + 8c as a factor two times.
(7b + 8c)2 = (7b + 8c)(7b + 8c)
Next, multiply 7b times each term of 7b + 8c, then multiply 8c times each term
of 7b + 8c. Finally, combine like terms.
= 7b(7b + 8c) + 8c(7b + 8c)
= 49b2 + 56bc + 56bc + 64c2
= 49b2 + 112bc + 64c2
Thus, (7b + 8c)2 = 49b2 + 112bc + 64c2 .
47. Multiply 2t2 times each term of 2t2 + 9t + 4, multiply 9t times each term
of 2t2 + 9t + 4, and multiply 4 times each term of 2t2 + 9t + 4. Finally, combine
like terms.
(2t2 + 9t + 4)(2t2 + 9t + 4)
= 2t2 (2t2 + 9t + 4) + 9t(2t2 + 9t + 4) + 4(2t2 + 9t + 4)
= 4t4 + 18t3 + 8t2 + 18t3 + 81t2 + 36t + 8t2 + 36t + 16
= 4t4 + 36t3 + 97t2 + 72t + 16
Thus, (2t2 + 9t + 4)(2t2 + 9t + 4) = 4t4 + 36t3 + 97t2 + 72t + 16.
49. Multiply 4w2 times each term of 3w2 − 6w + 8, multiply 3w times each
term of 3w2 − 6w + 8, and multiply 5 times each term of 3w2 − 6w + 8. Finally,
combine like terms.
(4w2 + 3w + 5)(3w2 − 6w + 8)
= 4w2 (3w2 − 6w + 8) + 3w(3w2 − 6w + 8) + 5(3w2 − 6w + 8)
= 12w4 − 24w3 + 32w2 + 9w3 − 18w2 + 24w + 15w2 − 30w + 40
= 12w4 − 15w3 + 29w2 − 6w + 40
Thus, (4w2 + 3w + 5)(3w2 − 6w + 8) = 12w4 − 15w3 + 29w2 − 6w + 40.
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51. To find the revenue, we must multiply the number of widgets sold by the
price for each widget. That is, we must multiply the demand times the unit
price.
Revenue = Demand × Unit Price
Let R represent the revenue. Because x represents the demand and p represents
the unit price, the last equation can be rewritten as follows.
R = xp
However, the demand is given by the equation x = 320 − 0.95p. Substitute
320 − 0.95p for x in the last equation to get
R = (320 − 0.95p)p,
or equivalently,
R = 320p − 0.95p2 .
We’re asked to find when the revenue equals R = 7, 804, so enter Y 1 =
320 ∗ X − 0.95 ∗ X ∧ 2 and Y 2 = 7804 into the Y= menu in your calculator,
then set the WINDOW parameters as follows: Xmin=0, Xmax=350, Ymin=0,
Ymax=27000. Push the GRAPH button to produce the graph, then use the
5:intersect utility to determine the coordinates of the points of intersection.
Report your answer on your homework as follows.
R(dollars)
27,000
R = 320p − 0.95p2
R = 7804
0
026.47
310.37350
p(dollars)
Hence, the revenue will equal R = $7, 804 if the unit price is set either at $26.47
or $310.37.
53. Because the edge of the outer square is 6 inches longer that 3 times the
edge of the inner square, the edge of the outer square is 3x + 6.
x
3x + 6
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To find the area of the shaded region, subtract the area of the smaller square
from the area of the larger square. Recall that the area of a square is found by
squaring the length of one of its sides.
Area of shaded region = Area of larger square − Area of smaller square
A(x) = (3x + 6)2 − x2
Use the distributive property to multiply.
A(x) = (3x + 6)(3x + 6) − x2
A(x) = 9x2 + 18x + 18x + 36 − x2
Combine like terms.
A(x) = 8x2 + 36x + 36
Hence, the area of the shaded region is given by the polynomial A(x) = 8x2 +
36x + 36. We can now evaluate the polynomial at x = 5 inches by substituting
5 for x.
A(x) = 8x2 + 36x + 36
A(5) = 8(5)2 + 36(5) + 36
Use your calculator to help simplify.
A(5) = 416
Hence, the area of the shaded region is 416 square inches.
55. If the width of the entire rectangular garden is 29 feet, and the width of
the border lawn is x feet, then the width of the interior rectangular garden
is 29 − 2x feet. Simlarly, if the length of the entire rectangular garden is 31
feet, and the width of the border lawn is x feet, then the length of the interior
rectangular garden is 31 − 2x feet.
29 − 2x
x
29
x
x
31 − 2x
x
31
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