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3.6e: Exercises - Zeroes of Polynomial Functions

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    45013
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    A: Concepts

    Exercise \(\PageIndex{A}\): Concepts

    1) Describe a use for the Remainder Theorem.

    2) Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.

    3) What is the difference between rational and real zeros?

    4) If Descartes’ Rule of Signs reveals a \(0\) or \(1\) change of signs, what specific conclusion can be drawn?

    5) If synthetic division reveals a zero, why should we try that value again as a possible solution?

    Answers to odd exercises: 

    1. The theorem can be used to evaluate a polynomial.

    3. Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

    5. Polynomials can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

    B: Use the Remainder Theorem to Evaluate a Polynomial

    Exercise \(\PageIndex{B}\): Use the Remainder Theorem

    \( \bigstar \) Use synthetic division to evaluate \(p(c)\) and write \(p(x)\) in the form \(p(x) = (x-c) q(x) +r\).

    6. \(p(x) = 2x^2 - x + 1\), \(c = 4\)

    7. \(p(x) = 4x^2-33x-180\), \(c = 12\)

    8. \(p(x) = 2x^3 - x + 6\), \(c=-3\)

    9. \(p(x) = x^3+2x^2+3x+4\), \(c =-1\)

    10. \(p(x) =3x^3-6x^2+4x-8\), \(c=2\)

    11. \(p(x) = 8x^3+12x^2+6x+1\), \(c =-\frac{1}{2}\)

    12. \(p(x) = 2x^4 +x^3- 4x^2+10x-7\), \(c=\frac{3}{2}\)

    13.  \(p(x) = x^4 - 3x^3 - 20x^2 - 24x - 8\), \(c =7\) 

    14. \(p(x) = x^4 - 5x^2 - 8x -12\), \(c=3\)

    15. \(p(x) = x^4 - 5x^3 + x^2 + 5\), \(c =2\)

    Answers to odd exercises:

    7. \(p(12) =0\), \(p(x) = (x-12)(4x+15) \)

    9. \(p(-1)=2\), \(p(x) = (x+1)(x^2 + x+2) + 2 \)

    11. \(p\left(-\frac{1}{2}\right) = 0\), \(p(x) = (2x+1)(4x^2+4x+1)\)

    13. \(p(7)=216\), \(p(x) = (x-7)(x^3+4x^2 +8 x+32) + 216 \)

    15. \(p(2)=-15\), \(p(x) = (x-2)(x^3-3x^2 -5x -10) -15 \)

    C: Given one zero or factor, find all Real Zeros, and factor a polynomial

    Exercise \(\PageIndex{C}\): Use the Factor Theorem given one zero or factor

    \( \bigstar \) Given a polynomial and one of its factors, find the rest of the real zeros and write the polynomial as a product of linear and irreducible quadratic factors.

    17) \(f(x)=2x^3+x^2−5x+2;\) Factor: \( ( x+2) \)

    18) \(f(x)=3x^3+x^2−20x+12;\) Factor: \( ( x+3)\)

    19) \(f(x)=2x^3+3x^2+x+6;\) Factor: \( (x+2)\)

    20) \(f(x)=−5x^3+16x^2−9;\) Factor: \( (x−3)\)

    21) \(f(x)=x^3+3x^2+4x+12;\) Factor: \( (x+3)\)

    22) \(f(x)=4x^3−7x+3;\) Factor: \( (x−1)\)

    23) \(f(x)=2x^3+5x^2−12x−30;\) Factor: \( (2x+5)\)

    24) \(f(x)=2x^3−9x^2+13x−6;\) Factor: \( (x−1) \)

    Answers to odd exercises:

    17. \(−2, 1, \frac{1}{2}\); \( f(x)=(x+2)(x-1)(2x-1) \)

    19. \(−2\); \( f(x)=(x+2)(2x^2-x+3) \)

    21. \(−3\); \( f(x)=(x+3)(x^2+4) \)

    23. \(−\frac{5}{2},\; \sqrt{6},\; −\sqrt{6}; \) \(f(x)=(2x+5)(x-\sqrt{6})(x+\sqrt{6})\)

    \( \bigstar \) Given a polynomial and \(c\), one of its zeros, find the rest of the real zeros and write the polynomial as a product of linear and irreducible quadratic factors. It is possible some factors are repeated.

    25. \(p(x)=x^{3} - 24x^{2} + 192x - 512, \;\; c = 8\)

    26. \(p(x)=3x^{3} + 4x^{2} - x - 2, \;\; c = \frac{2}{3}\)

    27. \(p(x)=2x^3-3x^2-11x+6, \;\; c=\frac{1}{2}\)

    28. \(p(x)=x^3+2x^2-3x-6, \;\; c = -2\)

    29. \(p(x)=2x^3-x^2-10x+5, \;\; c=\frac{1}{2}\)

    30. \(p(x)=4x^{4} - 28x^{3} + 61x^{2} - 42x + 9,\; c = \frac{1}{2}\) 

    31. \(p(x)=x^5+2x^4-12x^3-38x^2-37x-12,\) \(\; c=-1\) 

    32. \(p(x)=2x^5 +7x^4 - 18x^2 - 8x +8,\)  \(\; c = \frac{1}{2}\)

    33. \(p(x)=3x^5 +2x^4 - 15x^3 -10x^2 +12x +8,\)  \(\; c = -\frac{2}{3}\)

    Answers to odd exercises:

    25. zeros: \(8\);  \(p(x)= (x - 8)^{3}\)

    27. zeros: \( \frac{1}{2}, -2, 3 \);  \(p(x)= (2x-1)(x+2)(x-3)\)

    29. zeros: \( \frac{1}{2}, \pm \sqrt{5} \);  \(p(x)= (2x-1)(x+\sqrt{5})(x-\sqrt{5})\)

    31. zeros: \( -1,\) \(-3,\) \(4\);  \(p(x)= (x+1)^3(x+3)(x-4)\)

    33. zeros: \( -2,\; -1,\; -\frac{2}{3},\; 1,\; 2 \\ \);
    \( \quad\)  \(p(x)= (x+2)(x+1)(x-1)(x-2)(3x+2)\)

    D: List all Possible Rational Zeros

    Exercise \(\PageIndex{D}\): Use the Rational Zero Theorem

    \( \bigstar \) Use the Rational Zeros Theorem to list all possible rational zeros for each given function. 

    35) \(f(x)=2x^3+3x^2−8x+5\)

    36) \(f(x)=3x^3+5x^2−5x+4\)

    37) \(f(x)=6x^4−10x^2+13x+1\)

    38) \(f(x)=4x^5−10x^4+8x^3+x^2−8\)

    39. \(f(x) = x^{3} - 2x^{2} - 5x + 6\)

    40. \(f(x) = x^{4} + 2x^{3} - 12x^{2} - 40x - 32\)

    41. \(f(x) = x^{3} - 7x^{2} + x - 7\)

    42. \(f(x) = x^{3} + 4x^{2} - 11x + 6\)

    43. \(f(x) = x^{4} - 9x^{2} - 4x + 12\)

    44. \(f(x) = -2x^{3} + 19x^{2} - 49x + 20\)

    45. \(f(x) = -17x^{3} + 5x^{2} + 34x - 10\)

    46. \(f(x) = 36x^{4} - 12x^{3} - 11x^{2} + 2x + 1\)

    47. \(f(x) = 3x^{3} + 3x^{2} - 11x - 10\)

    48. \(f(x) = 2x^4+x^3-7x^2-3x+3\)

    Answers to odd exercises:

    35. \(±5, ±1, ± \frac{1}{2}, ± \frac{5}{2}\)

    37. \(±1, ±\frac{1}{2}, ±\frac{1}{3}, ±\frac{1}{6}\)

    39. \(\pm 1\), \(\pm 2\), \(\pm 3\), \(\pm 6\) \(\qquad \qquad\) 41. \(\pm 1\), \(\pm 7\)

    43. \(\pm 1\), \(\pm 2\), \(\pm 3\), \(\pm 4\), \(\pm 6\), \(\pm 12\)

    45. \(\pm 1\), \(\pm 2\), \(\pm 5\), \(\pm 10\), \(\pm \frac{1}{17}\),\(\pm \frac{2}{17}\),\(\pm \frac{5}{17}\),\(\pm \frac{10}{17}\)

    47. \(\pm 1\), \(\pm 2\), \(\pm 5\), \(\pm 10\), \(\pm \frac{1}{3}\),\(\pm \frac{2}{3}\),\(\pm \frac{5}{3}\),\(\pm \frac{10}{3}\)

    E: Find all Zeros that are Rational

    Exercise \(\PageIndex{E}\): Find all zeros that are rational

    \( \bigstar \) Use the Rational Zero Theorem to find all real number zeros.

    49) \(x^3−3x^2−10x+24=0\)

    50) \(2x^3+7x^2−10x−24=0\)

    51) \(x^3+2x^2−9x−18=0\)

    52) \(x^3+5x^2−16x−80=0\)

    53) \(x^3−3x^2−25x+75=0\)

    54) \(2x^3−3x^2−32x−15=0\)

    55) \(2x^3+x^2−7x−6=0\)

    56) \(2x^3−3x^2−x+1=0\)

    57) \(3x^3−x^2−11x−6=0\)

    58) \(x^4−2x^3−7x^2+8x+12=0\)

    59) \(4x^3−3x+1=0\)

    60) \(4x^4+4x^3−25x^2−x+6=0\)

    61) \(x^4+2x^3−9x^2−2x+8=0\)

    62) \(x^4+2x^3−4x^2−10x−5=0\)

    63) \(-5x^4+4x^3−19x^2+16x+4=0\)

    Answers to odd exercises:
    49. \(-3,\; 2,\; 4\)

    51. \(-2,\; 3,\; −3\)

    53. \(3, −5, 5 \)

    55. \(2,\; -1,\; -\frac{3}{2}\)

    57. \( -\frac{2}{3} ,\; \frac{1 \pm \sqrt{13}}{2} \)

    59. \( -1,\; \frac{1}{2} \)

    61. \(1,\; 2,\; −1,\; −4\)

    63. \(−\frac{1}{5},\; 1  \)

    \( \bigstar \) Find the real zeros of the polynomial. State the multiplicity of each real zero.

    65. \(f(x) = x^{3} - 2x^{2} - 5x + 6\)

    66. \(f(x) = x^{3} + 4x^{2} - 11x + 6\)

    67. \(f(x) = x^{4} - 9x^{2} - 4x + 12\)

    68. \(f(x) = -17x^{3} + 5x^{2} + 34x - 10\)

    69. \(f(x) = 36x^{4} - 12x^{3} - 11x^{2} + 2x + 1\)

    70. \(f(x) = 2x^4+x^3-7x^2-3x+3\)

    71. \(f(x) = 2x^{3} + 7x^{2} +4x - 4\)

    72. \(f(x) = -2x^4 - 3x^3 +10x^2 + 12x - 8\)

    Answers to odd exercises:

    65. \(x = 1\) (mult. 1), \(x = 3\) (mult. 1), \(x = -2\) (mult. 1)

    67. \(x = -2\) (mult. 2), \(x = 1\) (mult. 1), \(x = 3\) (mult. 1)

    69. \(x = \frac{1}{2}\) (mult. 2), \( x = -\frac{1}{3}\) (mult. 2)

    71.  \(x = -2\) (mult. 2), \(x = \frac{1}{2}\) (mult. 1)

    F: Find all zeros (both real and imaginary)

    Exercise \(\PageIndex{F}\): Find all zeros

    \( \bigstar \) Use the Rational Zero Theorem to find all complex solutions (real and non-real).

    72) \(x^3+x^2+x+1=0\)

    73) \(x^3−8x^2+25x−26=0\)

    74) \(x^3+13x^2+57x+85=0\)

    75) \(3x^3−4x^2+11x+10=0\)

    76) \(x^4+2x^3+22x^2+50x−75=0\)

    77) \(2x^3−3x^2+32x+17=0\)

    Answers to odd exercises:
    73. \(2, 3+2i, 3−2i\) 75. \(−\dfrac{2}{3}, 1+2i, 1−2i\) 77. \(−\dfrac{1}{2}, 1+4i, 1−4i\)

    G: Find all zeros and sketch

    Exercise \(\PageIndex{G}\): Find all zeros and sketch

    \( \bigstar \) Determine the end behaviour, all the real zeros, their multiplicity, and y-intercept. Sketch the function. (Use synthetic division to find a rational zero. Use the quotient to find the next zero).

    78) \(f(x)=x^3−1\)

    79) \(f(x)=x^4−x^2−1\)

    80) \(f(x)=x^3−2x^2−5x+6\)

    81) \(f(x)=2x^3+37x^2+200x+300\) 

    82) \(f(x)=x^4+2x^3−12x^2+14x−5\)

    83) \(f(x)=2x^4−5x^3−5x^2+5x+3\)

    84) \(f(x)=x^3−2x^2−16x+32\)

    85) \(f(x)=-x^3−7x^2-8x+16\)

    86. \(f(x) = -x^4 - 4x^3+3x^2 +10x -8\)

    87. \(f(x) =  x^{4} - 6x^{3} + 8x^{2} + 6x - 9\)

    88. \(f(x) = x^{4} + 4x^{3} - 5x^{2} - 36x - 36\)

    89. \(f(x) = x^{5} - x^{4} - 5x^{3} + x^{2} + 8x + 4\)

    90. \(f(x) = x^{4} + 2x^{3} + 6x - 9\)

    Answers to odd exercises:

    79. zeros (odd multiplicity): \( \pm \sqrt{ \frac{1+\sqrt{5} }{2} }\), 2 imaginary zeros, y-intercept \( (0, 1) \)

          CNX_PreCalc_Figure_03_06_202.jpg

    81. zeros (odd multiplicity): \( \{-10, -6,  \frac{-5}{2} \} \); y-intercept: \( (0, 300) \)

      CNX_PreCalc_Figure_03_06_206 (1).jpg

    83. zeros (odd multiplicity); \( \{ -1, 1, 3, \frac{-1}{2} \} \), y-intercept \( (0,3) \). 

          CNX_PreCalc_Figure_03_06_208.jpg

    85. zeros; \(-4\) (multiplicity \(2\)), \(1\) (multiplicity \(1\)), y-intercept \( (0,16) \).   

    3.6E #85.png

    87.  odd multiplicity zeros: \( \{1, -1\} \); even multiplicity zero: \( \{ 3 \} \); y-intercept \( (0, -9) \). 

           3.6E #87.png

    89. odd multiplicity zero: \( \{ -1 \} \), even multiplicity zero \( \{ 2 \} \). y-intercept \( (0, 4) \). 

          3.6E #89.png

    H: Given zeros, construct a polynomial function

    Exercise \(\PageIndex{H}\): Given zeros, construct a polynomial function

    \( \bigstar \) Construct a polynomial function of least degree possible using the given information. You may leave the polynomial in factored form.

    91) A lowest degree polynomial with real coefficients and zero \( 3i \) 

    92) A lowest degree polynomial with rational coefficients and zeros: \( 2 \) and \( \sqrt{6} \)

    93) A lowest degree polynomial with integer coefficients and Real roots: \(–1\) (with multiplicity \(2\)), and \(1\).

    94) A lowest degree polynomial with integer coefficients and Real roots: \(–2\), and \(\frac{1}{2}\) (with multiplicity \(2\)) 

    95) A lowest degree polynomial with integer coefficients and Real roots:\(−\frac{1}{2}, 0,\frac{1}{2}\) 

    96) A lowest degree polynomial with integer coefficients and Real roots: \(–4, –1, 1, 4\) 

    97) A lowest degree polynomial with integer coefficients and Real roots: \(–1, 1, 3\) 

    98. A lowest degree polynomial with real coefficients and zeros: \(-2 \) and \( -5i \)

    99. A lowest degree polynomial with real coefficients and zeros: \(4 \) and \( 2i \).

    100. The solutions to \(p(x) = 0\) are \(x = \pm 3\) and \(x=6\).  The leading term of \(p(x)\) is \(7x^4\). 
    \(\qquad\)The point \((-3,0)\) is a local minimum on the graph of \(y=p(x)\).

    101. The solutions to \(p(x) =0\) are \(x = \pm 3\), \(x=-2\), and \(x=4\), The leading term of \(p(x)\) is \(-x^5\). 
    \(\qquad\)The point \((-2, 0)\) is a local maximum on the graph of \(y=p(x)\).

    102. \(p\) is degree 4. as \(x \rightarrow \infty\), \(p(x) \rightarrow -\infty\) \(p\) has exactly three \(x\)-intercepts: \((-6,0)\), \((1,0)\) and \((117,0)\). 
    \(\qquad\)The graph of \(y=p(x)\) crosses through the \(x\)-axis at \((1,0)\).

    103. Find a quadratic polynomial with integer coefficients which has \(x = \dfrac{3}{5} \pm \dfrac{\sqrt{29}}{5}\) as its real zeros.

    Answers to odd exercises:

    91. \(f(x)=(x^2+9)\)

    93. \(f(x)=(x+1)^2(x-1)\)

    95. \(f(x)=x(2x+1)(2x-1)\)

    97. \( f(x) = (x+1)(x-1)(x-3) \)

    99. \(p(x)= (x-4)(x-2i)(x+2i)=x^3-4x^2+4x-16\)

    101. \(p(x) = -(x + 2)^{2}(x - 3)(x + 3)(x - 4)\)

    103. \(p(x) = 5x^{2} - 6x - 4\)

    I: Use Intermediate Value Theorem

    Exercise \(\PageIndex{I}\): Intermediate Value Theorem

    \( \bigstar \) Use the Intermediate Value Theorem to confirm the polynomial \(f\) has at least one zero within the given interval.

    104) \(f(x)=x^3−9x\), between \(x=−4\) and \(x=−2\).

    105) \(f(x)=x^3−9x\), between \(x=2\) and \(x=4\).

    106) \(f(x)=x^5−2x\), between \(x=1\) and \(x=2\).

    107) \(f(x)=−x^4+4\), between \(x=1\) and \(x=3\).

    108) \(f(x)=−2x^3−x\), between \(x=–1\) and \(x=1\).

    109) \(f(x)=x^3−100x+2\), between \(x=0.01\) and \(x=0.1\)

    Answers to odd exercises:
    105. \(f(2)=–10,\; f(4)=28\).
    \(\qquad\)Sign change confirms.
    107. \(f(1)=3,\; f(3)=–77.\)
    \(\qquad\)Sign change confirms.
    109. \(f(0.01)=1.000001,\; f(0.1)=–7.999\).
    \(\qquad\)Sign change confirms.

    x


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