Practice Makes Perfect
Determine the Type of Polynomials
In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Also give the degree of each polynomial.
1. ⓐ \(47x^5−17x^2y^3+y^2\)
ⓑ \(5c^3+11c^2−c−8\)
ⓒ \(59ab+13b\)
ⓓ \(4\)
ⓔ \(4pq+17\)
- Answer
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ⓐ trinomial, degree 5
ⓑ other polynomial, degree 3
ⓒ binomial, degree 2
ⓓ monomial, degree 0
ⓔ binomial, degree 2
2. ⓐ \(x^2−y^2\)
ⓑ \(−13c^4\)
ⓒ \(a^2+2ab−7b^2\)
ⓓ \(4x^2y^2−3xy+8\)
ⓔ \(19\)
3. ⓐ \(8y−5x\)
ⓑ \(y^2−5yz−6z^2\)
ⓒ \(y^3−8y^2+2y−16\)
ⓓ \(81ab^4−24a^2b^2+3b\)
ⓔ \(−18\)
- Answer
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ⓐ binomial, degree 1
ⓑ trinomial, degree 2
ⓒ other polynomial, degree 3
ⓓ trinomial, degree 5
ⓔ monomial, degree 0
4. ⓐ \(11y^2\)
ⓑ \(−73\)
ⓒ \(6x^2−3xy+4x−2y+y^2\)
ⓓ \(4y^2+17z^2\)
ⓔ \(5c^3+11c^2−c−8\)
5. ⓐ \(5a^2+12ab−7b^2\)
ⓑ \(18xy^2z\)
ⓒ \(5x+2\)
ⓓ \(y^3−8y^2+2y−16\)
ⓔ \(−24\)
- Answer
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ⓐ trinomial, degree 2
ⓑ monomial, degree 4
ⓒ binomial, degree 1
ⓓ other polynomial, degree 3
ⓔ monomial, degree 0
6. ⓐ \(9y^3−10y^2+2y−6\)
ⓑ \(−12p^3q\)
ⓒ \(a^2+9ab+18b^2\)
ⓓ \(20x^2y^2−10a^2b^2+30\)
ⓔ \(17\)
7. ⓐ \(14s−29t\)
ⓑ \(z^2−5z−6\)
ⓒ \(y^3−8y^2z+2yz^2−16z^3\)
ⓓ \(23ab^2−14\)
ⓔ \(−3\)
- Answer
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ⓐ binomial, degree 1
ⓑ trinomial, degree 2
ⓒ other polynomial, degree 3
ⓓ binomial, degree 3
ⓔ monomial, degree 0
8. ⓐ \(15xy\)
ⓑ \(15\)
ⓒ \(6x^2−3xy+4x−2y+y^2\)
ⓓ \(10p−9q\)
ⓔ \(m^4+4m^3+6m^2+4m+1\)
Add and Subtract Polynomials
In the following exercises, add or subtract the monomials.
9. ⓐ \(7x^2+5x^2\)
ⓑ \(4a−9a\)
- Answer
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ⓐ \(12x^2\) ⓑ \(−5a\)
10. ⓐ \(4y^3+6y^3\)
ⓑ \(−y−5y\)
11. ⓐ \(−12w+18w\)
ⓑ \(7x^2y−(−12x^2y)\)
- Answer
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ⓐ \(6w\)
ⓑ \(19x^2y\)
12. ⓐ \(−3m+9m\)
ⓑ \(15yz^2−(−8yz^2)\)
13. \(7x^2+5x^2+4a−9a\)
- Answer
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\(12x^2−5a\)
15. \(−12w+18w+7x^2y−(−12x^2y)\)
- Answer
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\(6w+19x^2y\)
16. \(−3m+9m+15yz^2−(−8yz^2)\)
17. ⓐ \(−5b−17b\)
ⓑ \(3xy−(−8xy)+5xy\)
- Answer
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ⓐ \(−22b\)
ⓑ \(16xy\)
18. ⓐ \(−10x−35x\)
ⓑ \(17mn^2−(−9mn^2)+3mn^2\)
19. ⓐ \(12a+5b−22a\)
ⓑ \(pq^2−4p−3q^2\)
- Answer
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ⓐ \(−10a+5b\)
ⓑ \(pq^2−4p−3q^2\)
20. ⓐ \(14x−3y−13x\)
ⓑ \(a^2b−4a−5ab^2\)
21. ⓐ \(2a^2+b^2−6a^2\)
ⓑ \(x^2y−3x+7xy^2\)
- Answer
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ⓐ \(−4a^2+b^2\)
ⓑ \(x^2y−3x+7xy^2\)
22. ⓐ \(5u^2+4v^2−6u^2\)
ⓑ \(12a+8b\)
23. ⓐ \(xy^2−5x−5y^2\)
ⓑ \(19y+5z\)
- Answer
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ⓐ \(xy^2−5x−5y^2\)
ⓑ \(19y+5z\)
24. \(12a+5b−22a+pq^2−4p−3q^2\)
25. \(14x−3y−13x+a^2b−4a−5ab^2\)
- Answer
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\(x−3y+a^2b−4a−5ab^2\)
26. \(2a^2+b^2−6a^2+x^2y−3x+7xy^2\)
27. \(5u^2+4v^2−6u^2+12a+8b\)
- Answer
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\(−u^2+4v^2+12a+8b\)
28. \(xy^2−5x−5y^2+19y+5z\)
29. Add: \(4a,−3b,−8a\)
- Answer
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\(−4a−3b\)
31. Subtract \(5x^6\) from \(−12x^6\)
- Answer
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\(−7x^6\)
32. Subtract \(2p^4\) from \(−7p^4\)
In the following exercises, add the polynomials.
33. \((5y^2+12y+4)+(6y^2−8y+7)\)
- Answer
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\(11y^2+4y+11\)
34. \((4y^2+10y+3)+(8y^2−6y+5)\)
35. \((x^2+6x+8)+(−4x^2+11x−9)\)
- Answer
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\(−3x^2+17x−1\)
36. \((y^2+9y+4)+(−2y^2−5y−1)\)
37. \((8x^2−5x+2)+(3x^2+3)\)
- Answer
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\(11x^2−5x+5\)
38. \((7x^2−9x+2)+(6x^2−4)\)
39. \((5a^2+8)+(a^2−4a−9)\)
- Answer
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\(6a^2−4a−1\)
40. \((p^2−6p−18)+(2p^2+11)\)
In the following exercises, subtract the polynomials.
41. \((4m^2−6m−3)−(2m^2+m−7)\)
- Answer
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\(2m^2−7m+4\)
42. \((3b^2−4b+1)−(5b^2−b−2)\)
43. \((a^2+8a+5)−(a^2−3a+2)\)
- Answer
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\(11a+3\)
44. \((b^2−7b+5)−(b^2−2b+9)\)
45. \((12s^2−15s)−(s−9)\)
- Answer
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\(12s^2−14s+9\)
46. \((10r^2−20r)−(r−8)\)
In the following exercises, subtract the polynomials.
47. Subtract \((9x^2+2)\) from \((12x^2−x+6)\)
- Answer
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\(3x^2−x+4\)
48. Subtract \((5y^2−y+12)\) from \((10y^2−8y−20)\)
49. Subtract \((7w^2−4w+2)\) from \((8w^2−w+6)\)
- Answer
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\(w^2+3w+4\)
50. Subtract \((5x^2−x+12)\) from \((9x^2−6x−20)\)
In the following exercises, find the difference of the polynomials.
51. Find the difference of \((w^2+w−42)\) and \((w^2−10w+24)\)
- Answer
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\(11w−64\)
52. Find the difference of \((z^2−3z−18)\) and \((z^2+5z−20)\)
In the following exercises, add the polynomials.
53. \((7x^2−2xy+6y^2)+(3x^2−5xy)\)
- Answer
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\(10x^2−7xy+6y^2\)
54. \((−5x^2−4xy−3y^2)+(2x^2−7xy)\)
55. \((7m^2+mn−8n^2)+(3m^2+2mn)\)
- Answer
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\(10m^2+3mn−8n^2\)
56. \((2r^2−3rs−2s^2)+(5r^2−3rs)\)
In the following exercises, add or subtract the polynomials.
57. \((a^2−b^2)−(a^2+3ab−4b^2)\)
- Answer
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\(−3ab+3b^2\)
58. \((m^2+2n^2)−(m^2−8mn−n^2)\)
59. \((p^3−3p^2q)+(2pq^2+4q^3)−(3p^2q+pq^2)\)
- Answer
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\(p^3−6p^2q+pq^2+4q^3\)
60. \((a^3−2a^2b)+(ab^2+b^3)−(3a^2b+4ab^2)\)
61. \((x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2)\)
- Answer
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\(x^3+2x^2y−5xy^2+y^3\)
62. \((x^3−2x^2y)−(xy^2−3y^3)−(x^2y−4xy^2)\)
Evaluate a Polynomial Function for a Given Value
In the following exercises, find the function values for each polynomial function.
63. For the function \(f(x)=8x^2−3x+2\), find:
ⓐ \(f(5)\) ⓑ \(f(−2)\) ⓒ \(f(0)\)
- Answer
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ⓐ \(187\) ⓑ \(40\) ⓒ \(2\)
64. For the function \(f(x)=5x^2−x−7\), find:
ⓐ \(f(−4)\) ⓑ \(f(1)\) ⓒ \(f(0)\)
65. For the function \(g(x)=4−36x\), find:
ⓐ \(g(3)\) ⓑ \(g(0)\) ⓒ \(g(−1)\)
- Answer
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ⓐ \(−104\) ⓑ \(4\) ⓒ \(40\)
66. For the function \(g(x)=16−36x^2\), find:
ⓐ \(g(−1)\) ⓑ \(g(0)\) ⓒ \(g(2)\)
In the following exercises, find the height for each polynomial function.
67. A painter drops a brush from a platform \(75\) feet high. The polynomial function \(h(t)=−16t^2+75\) gives the height of the brush \(t\) seconds after it was dropped. Find the height after \(t=2\) seconds.
- Answer
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The height is 11 feet.
68. A girl drops a ball off the cliff into the ocean. The polynomial \(h(t)=−16t^2+200\) gives the height of a ball \(t\) seconds after it is dropped. Find the height after \(t=3\) seconds.
69. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of \(p\) dollars each is given by the polynomial function \(R(p)=−4p^2+420p\). Find the revenue received when \(p=60\) dollars.
- Answer
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The revenue is $10,800.
70. A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of \(p\) dollars each is given by the polynomial \(R(p)=−4p^2+420p\). Find the revenue received when \(p=90\) dollars.
71. The polynomial \(C(x)=6x^2+90x\) gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side \(x\) feet and height \(6\) feet. Find the cost of producing a box with \(x=4\) feet.
- Answer
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The cost is $456.
72. The polynomial \(C(x)=6x^2+90x\) gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side \(x\) feet and height \(4\) feet. Find the cost of producing a box with \(x=6\) feet.
Add and Subtract Polynomial Functions
In each example, find ⓐ \((f+g)(x)\) ⓑ \((f+g)(2)\) ⓒ \((f-g)(x)\) ⓓ \((f-g)(3)\).
73. \(f(x)=2x^2−4x+1\) and \(g(x)=5x^2+8x+3\)
- Answer
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ⓐ \((f+g)(x)=7x^2+4x+4\)
ⓑ \((f+g)(2)=40\)
ⓒ \((f−g)(x)=−3x^2−12x−2\)
ⓓ \((f−g)(−3)=7\)
74. \(f(x)=4x^2−7x+3\) and \(g(x)=4x^2+2x−1\)
75. \(f(x)=3x^3−x^2−2x+3\) and \(g(x)=3x^3−7x\)
- Answer
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ⓐ \((f+g)(x)=6x^3−x^2−9x+3\)
ⓑ \((f+g)(2)=29\)
ⓒ \((f−g)(x)=−x^2+5x+3\)
ⓓ \((f−g)(−3)=−21\)
76. \(f(x)=5x^3−x^2+3x+4\) and \(g(x)=8x^3−1\)