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Mathematics LibreTexts

6.1E: Exercises

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    Practice Makes Perfect

    Identify Polynomials, Monomials, Binomials, and Trinomials

    In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

    Exercise 1
    1. \(81b^5−24b^3+1\)
    2. \(5c^3+11c^2−c−8\)
    3. \(\frac{14}{15}y+\frac{1}{7}\)
    4. \(5\)
    5. \(4y+17\)
    Answer
    1. trinomial
    2. polynomial
    3. binomial
    4. monomial
    5. binomial
    Exercise 2
    1. \(x^2−y^2\)
    2. \(−13c^4\)
    3. \(x^2+5x−7\)
    4. \(x^{2}y^2−2xy+8\)
    5. \(19\)
    Exercise 3
    1. \(8−3x\)
    2. \(z^2−5z−6\)
    3. \(y^3−8y^2+2y−16\)
    4. \(81b^5−24b^3+1\)
    5. \(−18\)
    Answer
    1. binomial
    2. trinomial
    3. polynomial
    4. trinomial
    5. monomial
    Exercise 4
    1. \(11y^2\)
    2. \(−73\)
    3. \(6x^2−3xy+4x−2y+y^2\)
    4. \(4y+17\)
    5. \(5c^3+11c^2−c−8\)

    Determine the Degree of Polynomials

    In the following exercises, determine the degree of each polynomial.

    Exercise 5
    1. \(6a^2+12a+14\)
    2. \(18xy^{2}z\)
    3. \(5x+2\)
    4. \(y^3−8y^2+2y−16\)
    5. \(−24\)
    Answer
    1. 2
    2. 4
    3. 1
    4. 3
    5. 0
    Exercise 6
    1. \(9y^3−10y^2+2y−6\)
    2. \(−12p^4\)
    3. \(a^2+9a+18\)
    4. \(20x^{2}y^2−10a^{2}b^2+30\)
    5. \(17\)
    Exercise 7
    1. \(14−29x\)
    2. \(z^2−5z−6\)
    3. \(y^3−8y^2+2y−16\)
    4. \(23ab^2−14\)
    5. \(−3\)
    Answer
    1. 1
    2. 2
    3. 3
    4. 3
    5. 0
    Exercise 8
    1. \(62y^2\)
    2. \(15\)
    3. \(6x^2−3xy+4x−2y+y^2\)
    4. \(10−9x\)
    5. \(m^4+4m^3+6m^2+4m+1\)
    Add and Subtract Monomials

    In the following exercises, add or subtract the monomials.

    Exercise 9

    \(7x^2+5x^2\)

    Answer

    \(12x^2\)

    Exercise 10

    \(4y^3+6y^3\)

    Exercise 11

    \(−12w+18w\)

    Answer

    \(6w\)

    Exercise 12

    \(−3m+9m\)

    Exercise 13

    \(4a−9a\)

    Answer

    \(−5a\)

    Exercise 14

    \(−y−5y\)

    Exercise 15

    \(28x−(−12x)\)

    Answer

    \(40x\)

    Exercise 16

    \(13z−(−4z)\)

    Exercise 17

    \(−5b−17b\)

    Answer

    \(−22b\)

    Exercise 18

    \(−10x−35x\)

    Exercise 19

    \(12a+5b−22a\)

    Answer

    \(−10a+5b\)

    Exercise 20

    \(14x−3y−13x\)

    Exercise 21

    \(2a^2+b^2−6a^2\)

    Answer

    \(−4a^2+b^2\)

    Exercise 22

    \(5u^2+4v^2−6u^2\)

    Exercise 23

    \(xy^2−5x−5y^2\)

    Answer

    \(xy^2−5x−5y^2\)

    Exercise 24

    \(pq^2−4p−3q^2\)

    Exercise 25

    \(a^{2}b−4a−5ab^2\)

    Answer

    \(a^{2}b−4a−5ab^2\)

    Exercise 26

    \(x^{2}y−3x+7xy^2\)

    Exercise 27

    \(12a+8b\)

    Answer

    \(12a+8b\)

    Exercise 28

    \(19y+5z\)

    Exercise 29

    Add: \(4a,\,−3b,\,−8a\)

    Answer

    \(−4a−3b\)

    Exercise 30

    Add: \(4x,\,3y,\,−3x\)

    Exercise 31

    Subtract \(5x^6\) from \(−12x^6\)

    Answer

    \(−17x^6\)

    Exercise 32

    Subtract \(2p^4\) from \(−7p^4\)

    ​​​​​​Add and Subtract Polynomials

    In the following exercises, add or subtract the polynomials.

    Exercise 33

    \((5y^2+12y+4)+(6y^2−8y+7)\)

    Answer

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

    Exercise 34

    \((4y^2+10y+3)+(8y^2−6y+5)\)

    Exercise 35

    \((x^2+6x+8)+(−4x^2+11x−9)\)

    Answer

    \(−3x^2+17x−1\)

    Exercise 36

    \((y^2+9y+4)+(−2y^2−5y−1)\)

    Exercise 37

    \((8x^2−5x+2)+(3x^2+3)\)

    Answer

    \(11x^2−5x+5\)

    Exercise 38

    \((7x^2−9x+2)+(6x^2−4)\)

    Exercise 39

    \((5a^2+8)+(a^2−4a−9)\)

    Answer

    \(6a^2−4a−1\)

    Exercise 40

    \((p^2−6p−18)+(2p^2+11)\)

    Exercise 41

    \((4m^2−6m−3)−(2m^2+m−7)\)

    Answer

    \(2m^2−7m+4\)

    Exercise 42

    \((3b^2−4b+1)−(5b^2−b−2)\)

    Exercise 43

    \((a^2+8a+5)−(a^2−3a+2)\)

    Answer

    \(11a+3\)

    Exercise 44

    \((b^2−7b+5)−(b^2−2b+9)\)

    Exercise 45

    \((12s^2−15s)−(s−9)\)

    Answer

    \(12s^2−16s+9\)

    Exercise 46

    \((10r^2−20r)−(r−8)\)

    Exercise 47

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

    Answer

    \(3x^2−x+4\)

    Exercise 48

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

    Exercise 49

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

    Answer

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

    Exercise 50

    Subtract \((5x^2−x+12)\) from \((9x^2−6x−20)\)

    Exercise 51

    Find the sum of \((2p^3−8)\) and \((p^2+9p+18)\)

    Answer

    \(2p^3+p^2+9p+10\)

    Exercise 52

    Find the sum of
    \((q^2+4q+13)\) and \((7q^3−3)\)

    Exercise 53

    Find the sum of \((8a^3−8a)\) and \((a^2+6a+12)\)

    Answer

    \(8a^3+a^2−2a+12\)

    Exercise 54

    Find the sum of
    \((b^2+5b+13)\) and \((4b^3−6)\)

    Exercise 55

    Find the difference of

    \((w^2+w−42)\) and
    \((w^2−10w+24)\).

    Answer

    \(11w−66\)

    Exercise 56

    Find the difference of
    \((z^2−3z−18)\) and
    \((z^2+5z−20)\)

    Exercise 57

    Find the difference of
    \((c^2+4c−33)\) and
    \((c^2−8c+12)\)

    Answer

    \(12c−45\)

    Exercise 58

    Find the difference of
    \((t^2−5t−15)\) and
    \((t^2+4t−17)\)

    Exercise 59

    \((7x^2−2xy+6y^2)+(3x^2−5xy)\)

    Answer

    \(10x^2−7xy+6y^2\)

    Exercise 60

    \((−5x^2−4xy−3y^2)+(2x^2−7xy)\)

    Exercise 61

    \((7m^2+mn−8n^2)+(3m^2+2mn)\)

    Answer

    \(10m^2+3mn−8n^2\)

    Exercise 62

    \((2r^2−3rs−2s^2)+(5r^2−3rs)\)

    Exercise 63

    \((a^2−b^2)−(a^2+3ab−4b^2)\)

    Answer

    \(−3ab+3b^2\)

    Exercise 64

    \((m^2+2n^2)−(m^2−8mn−n^2)\)

    Exercise 65

    \((u^2−v^2)−(u^2−4uv−3v^2)\)

    Answer

    \(4uv+2v^2\)

    Exercise 66

    \((j^2−k^2)−(j^2−8jk−5k^2)\)

    Exercise 67

    \((p^3−3p^{2}q)+(2pq^2+4q^3) −(3p^{2}q+pq^2)\)

    Answer

    \(p^3−6p^{2}q+pq^2+4q^3\)

    Exercise 68

    \((a^3−2a^{2}b)+(ab^2+b^3)−(3a^{2}b+4ab^2)\)

    Exercise 69

    \((x^3−x^{2}y)−(4xy^2−y^3)+(3x^{2}y−xy^2)\)

    Answer

    \(x^3+2x^{2}y−5xy^2+y^3\)

    Exercise 70

    \((x^3−2x^{2}y)−(xy^2−3y^3)−(x^{2}y−4xy^2)\)

    Evaluate a Polynomial for a Given Value

    In the following exercises, evaluate each polynomial for the given value.

    Exercise 71

    Evaluate \(8y^2−3y+2\) when:

    1. \(y=5\)
    2. \(y=−2\)
    3. \(y=0\)
    Answer
    1. \(187\)
    2. \(46\)
    3. \(2\)
    Exercise 72

    Evaluate \(5y^2−y−7\) when:

    1. \(y=−4\)
    2. \(y=1\)
    3. \(y=0\)
    Exercise 73

    Evaluate \(4−36x\) when:

    1. \(x=3\)
    2. \(x=0\)
    3. \(x=−1\)
    Answer
    1. \(−104\)
    2. \(4\)
    3. \(40\)
    Exercise 74

    Evaluate \(16−36x^2\) when:

    1. \(x=−1\)
    2. \(x=0\)
    3. \(x=2\)
    Exercise 75

    A painter drops a brush from a platform \(75\) feet high. The polynomial \(−16t^2+75\) gives the height of the brush \(t\) seconds after it was dropped. Find the height after \(t=2\) seconds.

    Answer

    \(11\)

    Exercise 76

    A girl drops a ball off a cliff into the ocean. The polynomial \(−16t^2+250\) gives the height of a ball \(t\) seconds after it is dropped from a 250-foot tall cliff. Find the height after \(t=2\) seconds.

    Exercise 77

    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 \(−4p^2+420p\). Find the revenue received when \(p=60\) dollars.

    Answer

    \($10,800\)

    Exercise 78

    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 \(−4p^2+420p\). Find the revenue received when \(p=90\) dollars.

    Everyday Math

    Exercise 79

    Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of \(x\) miles per hour is given by the polynomial \(−\frac{1}{150}x^2+\frac{1}{3}x\), where \(x=30\) mph.

    Answer

    \(4\)

    Exercise 80

    Stopping Distance The number of feet it takes for a car traveling at \(x\) miles per hour to stop on dry, level concrete is given by the polynomial \(0.06x^2+1.1x\), where \(x=40\) mph.

    Exercise 81

    Rental Cost The cost to rent a rug cleaner for \(d\) days is given by the polynomial \(5.50d+25\). Find the cost to rent the cleaner for \(6\) days.

    Answer

    \($58\)

    Exercise 82

    Height of Projectile The height (in feet) of an object projected upward is given by the polynomial \(−16t^2+60t+90\) where \(t\) represents time in seconds. Find the height after \(t=2.5\) seconds.

    Exercise 83

    Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial \(\frac{9}{5}c+32\) where \(c\) represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when \(c=65°\).

    Answer

    \(149°\) F

    Writing Exercises

    Exercise 84

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

    Exercise 85

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

    Answer

    Answers will vary.

    Exercise 86

    Ariana thinks the sum \(6y^2+5y^4\) is \(11y^6\)

    Exercise 87

    Jonathan thinks that \(\frac{1}{3}\) and \(\frac{1}{x}\) are both monomials. What is wrong with his reasoning?

    Answer

    Answers will vary.

    Self Check

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

    This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “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.” The rest of the cells are blank.

    b. 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.