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Mathematics LibreTexts

5.1E: Exercises

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    30850
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    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

    ⓐ 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

    ⓐ 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

    ⓐ 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

    ⓐ 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

    ⓐ \(12x^2\) ⓑ \(−5a\)

    10. ⓐ \(4y^3+6y^3\)
    ⓑ \(−y−5y\)

    11. ⓐ \(−12w+18w\)
    ⓑ \(7x^2y−(−12x^2y)\)

    Answer

    ⓐ \(6w\)
    ⓑ \(19x^2y\)

    12. ⓐ \(−3m+9m\)
    ⓑ \(15yz^2−(−8yz^2)\)

    13. \(7x^2+5x^2+4a−9a\)

    Answer

    \(12x^2−5a\)

    14. \(4y^3+6y^3−y−5y\)

    15. \(−12w+18w+7x^2y−(−12x^2y)\)

    Answer

    \(6w+19x^2y\)

    16. \(−3m+9m+15yz^2−(−8yz^2)\)

    17. ⓐ \(−5b−17b\)
    ⓑ \(3xy−(−8xy)+5xy\)

    Answer

    ⓐ \(−22b\)
    ⓑ \(16xy\)

    18. ⓐ \(−10x−35x\)
    ⓑ \(17mn^2−(−9mn^2)+3mn^2\)

    19. ⓐ \(12a+5b−22a\)
    ⓑ \(pq^2−4p−3q^2\)

    Answer

    ⓐ \(−10a+5b\)
    ⓑ \(pq^2−4p−3q^2\)

    20. ⓐ \(14x−3y−13x\)
    ⓑ \(a^2b−4a−5ab^2\)

    21. ⓐ \(2a^2+b^2−6a^2\)
    ⓑ \(x^2y−3x+7xy^2\)

    Answer

    ⓐ \(−4a^2+b^2\)
    ⓑ \(x^2y−3x+7xy^2\)

    22. ⓐ \(5u^2+4v^2−6u^2\)
    ⓑ \(12a+8b\)

    23. ⓐ \(xy^2−5x−5y^2\)
    ⓑ \(19y+5z\)

    Answer

    ⓐ \(xy^2−5x−5y^2\)
    ⓑ \(19y+5z\)

    24. \(12a+5b−22a+pq^2−4p−3q^2\)

    25. \(14x−3y−13x+a^2b−4a−5ab^2\)

    Answer

    \(x−3y+a^2b−4a−5ab^2\)

    26. \(2a^2+b^2−6a^2+x^2y−3x+7xy^2\)

    27. \(5u^2+4v^2−6u^2+12a+8b\)

    Answer

    \(−u^2+4v^2+12a+8b\)

    28. \(xy^2−5x−5y^2+19y+5z\)

    29. Add: \(4a,−3b,−8a\)

    Answer

    \(−4a−3b\)

    30. Add: \(4x,3y,−3x\)

    31. Subtract \(5x^6\) from \(−12x^6\)

    Answer

    \(−7x^6\)

    32. Subtract \(2p^4\) from \(−7p^4\)

    In the following exercises, add the polynomials.

    33. \((5y^2+12y+4)+(6y^2−8y+7)\)

    Answer

    \(11y^2+4y+11\)

    34. \((4y^2+10y+3)+(8y^2−6y+5)\)

    35. \((x^2+6x+8)+(−4x^2+11x−9)\)

    Answer

    \(−3x^2+17x−1\)

    36. \((y^2+9y+4)+(−2y^2−5y−1)\)

    37. \((8x^2−5x+2)+(3x^2+3)\)

    Answer

    \(11x^2−5x+5\)

    38. \((7x^2−9x+2)+(6x^2−4)\)

    39. \((5a^2+8)+(a^2−4a−9)\)

    Answer

    \(6a^2−4a−1\)

    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

    \(2m^2−7m+4\)

    42. \((3b^2−4b+1)−(5b^2−b−2)\)

    43. \((a^2+8a+5)−(a^2−3a+2)\)

    Answer

    \(11a+3\)

    44. \((b^2−7b+5)−(b^2−2b+9)\)

    45. \((12s^2−15s)−(s−9)\)

    Answer

    \(12s^2−14s+9\)

    46. \((10r^2−20r)−(r−8)\)

    In the following exercises, subtract the polynomials.

    47. Subtract \((9x^2+2)\) from \((12x^2−x+6)\)

    Answer

    \(3x^2−x+4\)

    48. Subtract \((5y^2−y+12)\) from \((10y^2−8y−20)\)

    49. Subtract \((7w^2−4w+2)\) from \((8w^2−w+6)\)

    Answer

    \(w^2+3w+4\)

    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

    \(11w−64\)

    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

    \(10x^2−7xy+6y^2\)

    54. \((−5x^2−4xy−3y^2)+(2x^2−7xy)\)

    55. \((7m^2+mn−8n^2)+(3m^2+2mn)\)

    Answer

    \(10m^2+3mn−8n^2\)

    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

    \(−3ab+3b^2\)

    58. \((m^2+2n^2)−(m^2−8mn−n^2)\)

    59. \((p^3−3p^2q)+(2pq^2+4q^3)−(3p^2q+pq^2)\)

    Answer

    \(p^3−6p^2q+pq^2+4q^3\)

    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

    \(x^3+2x^2y−5xy^2+y^3\)

    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

    ⓐ \(187\) ⓑ \(40\) ⓒ \(2\)

    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

    ⓐ \(−104\) ⓑ \(4\) ⓒ \(40\)

    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

    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

    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

    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

    ⓐ \((f+g)(x)=7x^2+4x+4\)
    ⓑ \((f+g)(2)=40\)
    ⓒ \((f−g)(x)=−3x^2−12x−2\)
    ⓓ \((f−g)(−3)=7\)

    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

    ⓐ \((f+g)(x)=6x^3−x^2−9x+3\)
    ⓑ \((f+g)(2)=29\)
    ⓒ \((f−g)(x)=−x^2+5x+3\)
    ⓓ \((f−g)(−3)=−21\)

    76. \(f(x)=5x^3−x^2+3x+4\) and \(g(x)=8x^3−1\)

    Writing Exercises

    77. Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

    Answer

    Answers will vary.

    78. Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of \(5\).

    79. Ariana thinks the sum \(6y^2+5y^4\) is \(11y^6\). What is wrong with her reasoning?

    Answer

    Answers will vary.

    80. Is every trinomial a second-degree polynomial? If not, give an example.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    The figure shows a table with six rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “identify polynomials, monomials, binomials, and trinomials”, “determine the degree of polynomials”, “add and subtract monomials”, “add and subtract polynomials”, and “evaluate a polynomial for a given value”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.

    ⓑ If most of your checks were:

    …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

    …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

    …no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.