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3.2E: Exercises

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    Exercises

    In Exercises 1 - 6, use polynomial long division to perform the indicated division. Write the polynomial in the form \(p(x) = d(x)q(x) + r(x)\).

    1. \(\left(4 x^{2}+3 x-1\right) \div(x-3)\)
    2. \(\left(2 x^{3}-x+1\right) \div\left(x^{2}+x+1\right)\)
    3. \(\left(5 x^{4}-3 x^{3}+2 x^{2}-1\right) \div\left(x^{2}+4\right)\)
    4. \(\left(-x^{5}+7 x^{3}-x\right) \div\left(x^{3}-x^{2}+1\right)\)
    5. \(\left(9 x^{3}+5\right) \div(2 x-3)\)
    6. \(\left(4 x^{2}-x-23\right) \div\left(x^{2}-1\right)\)

    In Exercises 7 - 20 use synthetic division to perform the indicated division. Write the polynomial in the form \(p(x) = d(x)q(x)+r(x)\).

    1. \(\left(3 x^{2}-2 x+1\right) \div(x-1)\)
    2. \(\left(x^{2}-5\right) \div(x-5)\)
    3. \(\left(3-4 x-2 x^{2}\right) \div(x+1)\)
    4. \(\left(4 x^{2}-5 x+3\right) \div(x+3)\)
    5. \(\left(x^{3}+8\right) \div(x+2)\)
    6. \(\left(4 x^{3}+2 x-3\right) \div(x-3)\)
    7. \(\left(18 x^{2}-15 x-25\right) \div\left(x-\frac{5}{3}\right)\)
    8. \(\left(4 x^{2}-1\right) \div\left(x-\frac{1}{2}\right)\)
    9. \(\left(2 x^{3}+x^{2}+2 x+1\right) \div\left(x+\frac{1}{2}\right)\)
    10. \(\left(3 x^{3}-x+4\right) \div\left(x-\frac{2}{3}\right)\)
    11. \(\left(2 x^{3}-3 x+1\right) \div\left(x-\frac{1}{2}\right)\)
    12. \(\left(4 x^{4}-12 x^{3}+13 x^{2}-12 x+9\right) \div\left(x-\frac{3}{2}\right)\)
    13. \(\left(x^{4}-6 x^{2}+9\right) \div(x-\sqrt{3})\)
    14. \(\left(x^{6}-6 x^{4}+12 x^{2}-8\right) \div(x+\sqrt{2})\)

    In Exercises 21 - 30, determine \(p(c)\) using the Remainder Theorem for the given polynomial functions and value of \(c\). If \(p(c)=0\), factor \(p(x)=(x-c) q(x)\).

    1. \(p(x)=2 x^{2}-x+1, c=4\)
    2. \(p(x)=4 x^{2}-33 x-180, c=12\)
    3. \(p(x)=2 x^{3}-x+6, c=-3\)
    4. \(p(x)=x^{3}+2 x^{2}+3 x+4, c=-1\)
    5. \(p(x)=3 x^{3}-6 x^{2}+4 x-8, c=2\)
    6. \(p(x)=8 x^{3}+12 x^{2}+6 x+1, c=-\frac{1}{2}\)
    7. \(p(x)=x^{4}-2 x^{2}+4, c=\frac{3}{2}\)
    8. \(p(x)=6 x^{4}-x^{2}+2, c=-\frac{2}{3}\)
    9. \(p(x)=x^{4}+x^{3}-6 x^{2}-7 x-7, c=-\sqrt{7}\)
    10. \(p(x)=x^{2}-4 x+1, c=2-\sqrt{3}\)

    In Exercises 31 - 40, you are given a polynomial and one of its zeros. Use the techniques in this section to find the rest of the real zeros and factor the polynomial.

    1. \(x^{3}-6 x^{2}+11 x-6, \quad c=1\)
    2. \(x^{3}-24 x^{2}+192 x-512, \quad c=8\)
    3. \(3 x^{3}+4 x^{2}-x-2, \quad c=\frac{2}{3}\)
    4. \(2 x^{3}-3 x^{2}-11 x+6, \quad c=\frac{1}{2}\)
    5. \(x^{3}+2 x^{2}-3 x-6, \quad c=-2\)
    6. \(2 x^{3}-x^{2}-10 x+5, \quad c=\frac{1}{2}\)
    7. \(4 x^{4}-28 x^{3}+61 x^{2}-42 x+9, c=\frac{1}{2} \text { is a zero of multiplicity } 2\)
    8. \(x^{5}+2 x^{4}-12 x^{3}-38 x^{2}-37 x-12, c=-1 \text { is a zero of multiplicity } 3\)
    9. \(125 x^{5}-275 x^{4}-2265 x^{3}-3213 x^{2}-1728 x-324, c=-\frac{3}{5} \text { is a zero of multiplicity } 3\)
    10. \(x^{2}-2 x-2, \quad c=1-\sqrt{3}\)

    In Exercises 41 - 45, create a polynomial \(p\) which has the desired characteristics. You may leave the polynomial in factored form.

      • The zeros of \(p\) are \(c=\pm 2\) and \(c=\pm 1\)
      • The leading term of \(p(x)\) is \(117 x^{4}\).
      • The zeros of \(p\) are \(c = 1\) and \(c = 3\)
      • \(c = 3\) is a zero of multiplicity 2.
      • The leading term of \(p(x)\) is \(-5 x^{3}\)
      • The solutions to \(p(x) = 0\) are \(x=\pm 3\) and \(x = 6\)
      • The leading term of \(p(x)\) is \(7 x^{4}\)
      • The point (−3, 0) is a local minimum on the graph of \(y = p(x)\).
      • The solutions to \(p(x) = 0\) are \(x=\pm 3\), \(x = −2\), and \(x = 4\).
      • The leading term of \(p(x)\) is \(-x^{5}\).
      • The point (−2, 0) is a local maximum on the graph of \(y = p(x)\).
      • \(p\) is degree 4.
      • as \(x \rightarrow \infty\), \(p(x) \rightarrow-\infty\)
      • \(p\) has exactly three \(x\)-intercepts: (−6, 0), (1, 0) and (117, 0)
      • The graph of \(y = p(x)\) crosses through the x-axis at (1, 0).
    6. Find a quadratic polynomial with integer coefficients which has \(x=\frac{3}{5} \pm \frac{\sqrt{29}}{5}\) as its real zeros.

    Answers

    1. \(4 x^{2}+3 x-1=(x-3)(4 x+15)+44\)
    2. \(2 x^{3}-x+1=\left(x^{2}+x+1\right)(2 x-2)+(-x+3)\)
    3. \(5 x^{4}-3 x^{3}+2 x^{2}-1=\left(x^{2}+4\right)\left(5 x^{2}-3 x-18\right)+(12 x+71)\)
    4. \(-x^{5}+7 x^{3}-x=\left(x^{3}-x^{2}+1\right)\left(-x^{2}-x+6\right)+\left(7 x^{2}-6\right)\)
    5. \(9 x^{3}+5=(2 x-3)\left(\frac{9}{2} x^{2}+\frac{27}{4} x+\frac{81}{8}\right)+\frac{283}{8}\)
    6. \(4 x^{2}-x-23=\left(x^{2}-1\right)(4)+(-x-19)\)
    7. \(\left(3 x^{2}-2 x+1\right)=(x-1)(3 x+1)+2\)
    8. \(\left(x^{2}-5\right)=(x-5)(x+5)+20\)
    9. \(\left(3-4 x-2 x^{2}\right)=(x+1)(-2 x-2)+5\)
    10. \(\left(4 x^{2}-5 x+3\right)=(x+3)(4 x-17)+54\)
    11. \(\left(x^{3}+8\right)=(x+2)\left(x^{2}-2 x+4\right)+0\)
    12. \(\left(4 x^{3}+2 x-3\right)=(x-3)\left(4 x^{2}+12 x+38\right)+111\)
    13. \(\left(18 x^{2}-15 x-25\right)=\left(x-\frac{5}{3}\right)(18 x+15)+0\)
    14. \(\left(4 x^{2}-1\right)=\left(x-\frac{1}{2}\right)(4 x+2)+0\)
    15. \(\left(2 x^{3}+x^{2}+2 x+1\right)=\left(x+\frac{1}{2}\right)\left(2 x^{2}+2\right)+0\)
    16. \(\left(3 x^{3}-x+4\right)=\left(x-\frac{2}{3}\right)\left(3 x^{2}+2 x+\frac{1}{3}\right)+\frac{38}{9}\)
    17. \(\left(2 x^{3}-3 x+1\right)=\left(x-\frac{1}{2}\right)\left(2 x^{2}+x-\frac{5}{2}\right)-\frac{1}{4}\)
    18. \(\left(4 x^{4}-12 x^{3}+13 x^{2}-12 x+9\right)=\left(x-\frac{3}{2}\right)\left(4 x^{3}-6 x^{2}+4 x-6\right)+0\)
    19. \(\left(x^{4}-6 x^{2}+9\right)=(x-\sqrt{3})\left(x^{3}+\sqrt{3} x^{2}-3 x-3 \sqrt{3}\right)+0\)
    20. \(\left(x^{6}-6 x^{4}+12 x^{2}-8\right)=(x+\sqrt{2})\left(x^{5}-\sqrt{2} x^{4}-4 x^{3}+4 \sqrt{2} x^{2}+4 x-4 \sqrt{2}\right)+0\)
    21. \(p(4)=29\)
    22. \(p(12)=0, p(x)=(x-12)(4 x+15)\)
    23. \(p(-3)=-45\)
    24. \(p(-1)=2\)
    25. \(p(2)=0, p(x)=(x-2)\left(3 x^{2}+4\right)\)
    26. \(p\left(-\frac{1}{2}\right)=0, p(x)=\left(x+\frac{1}{2}\right)\left(8 x^{2}+8 x+2\right)\)
    27. \(p\left(\frac{3}{2}\right)=\frac{73}{16}\)
    28. \(p\left(-\frac{2}{3}\right)=\frac{74}{27}\)
    29. \(p(-\sqrt{7})=0, p(x)=(x+\sqrt{7})\left(x^{3}+(1-\sqrt{7}) x^{2}+(1-\sqrt{7}) x-\sqrt{7}\right)\)
    30. \(p(2-\sqrt{3})=0, p(x)=(x-(2-\sqrt{3}))(x-(2+\sqrt{3}))\)
    31. \(x^{3}-6 x^{2}+11 x-6=(x-1)(x-2)(x-3)\)
    32. \(x^{3}-24 x^{2}+192 x-512=(x-8)^{3}\)
    33. \(3 x^{3}+4 x^{2}-x-2=3\left(x-\frac{2}{3}\right)(x+1)^{2}\)
    34. \(2 x^{3}-3 x^{2}-11 x+6=2\left(x-\frac{1}{2}\right)(x+2)(x-3)\)
    35. \(x^{3}+2 x^{2}-3 x-6=(x+2)(x+\sqrt{3})(x-\sqrt{3})\)
    36. \(2 x^{3}-x^{2}-10 x+5=2\left(x-\frac{1}{2}\right)(x+\sqrt{5})(x-\sqrt{5})\)
    37. \(4 x^{4}-28 x^{3}+61 x^{2}-42 x+9=4\left(x-\frac{1}{2}\right)^{2}(x-3)^{2}\)
    38. \(x^{5}+2 x^{4}-12 x^{3}-38 x^{2}-37 x-12=(x+1)^{3}(x+3)(x-4)\)
    39. \(125 x^{5}-275 x^{4}-2265 x^{3}-3213 x^{2}-1728 x-324=125\left(x+\frac{3}{5}\right)^{3}(x+2)(x-6)\)
    40. \(x^{2}-2 x-2=(x-(1-\sqrt{3}))(x-(1+\sqrt{3}))\)
    41. \(p(x)=117(x+2)(x-2)(x+1)(x-1)\)
    42. \(p(x)=-5(x-1)(x-3)^{2}\)
    43. \(p(x)=7(x+3)^{2}(x-3)(x-6)\)
    44. \(p(x)=-(x+2)^{2}(x-3)(x+3)(x-4)\)
    45. \(p(x)=a(x+6)^{2}(x-1)(x-117) \text { or } p(x)=a(x+6)(x-1)(x-117)^{2} \text { where } a \text { can be any negative real number}\)
    46. \(p(x)=5 x^{2}-6 x-4\)

    3.2E: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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