Power and Polynomial Functions: Get Stronger

Introduction to Power and Polynomial Functions

For the following exercises, identify the function as a power function, a polynomial function, or neither.

  1. [latex]f(x)=(x^2)^3[/latex]
  2. [latex]f(x)=\frac{x^2}{x^2−1}[/latex]
  3. [latex]f(x)=3^{x+1}[/latex]

For the following exercises, find the degree and leading coefficient for the given polynomial.

  1. [latex]7−2x^2[/latex]
  2. [latex]x(4−x^2)(2x+1)[/latex]

For the following exercises, determine the end behavior of the functions.

  1. [latex]f(x)=x^4[/latex]
  2. [latex]f(x)=−x^4[/latex]
  3. [latex]f(x)=−2x^4−3x^2+x−1[/latex]
  4. [latex]f(x)=x^2(2x^3-x+1)[/latex]

For the following exercises, find the intercepts of the functions.

  1. [latex]f(t)=2(t−1)(t+2)(t−3)[/latex]
  2. [latex]f(x)=x^4−16[/latex]
  3. [latex]f(x)=x(x^2−2x−8)[/latex]

For the following exercises, determine the least possible degree of the polynomial function shown.

  1. Graph of an odd-degree polynomial.
  2. Graph of an odd-degree polynomial.
  3. Graph of an odd-degree polynomial.
  4. Graph of an odd-degree polynomial.

For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

  1. Graph of an odd-degree polynomial.
  2. Graph of an even-degree polynomial.
  3. Graph of an odd-degree polynomial.
  4. Graph of an odd-degree polynomial.
  5. [latex]V(m)=(3+2m)^3[/latex], where [latex]m[/latex] is the number of minutes elapsed.
  6. [latex]V(x)=x(8-2x)^2[/latex], where [latex]x[/latex] is the length of the square’s side in inches.

Graphs of Polynomial Functions

For the following exercises, find the [latex]x[/latex]– or [latex]t[/latex]-intercepts of the polynomial functions.

  1. [latex]C(t)=3(t+2)(t−3)(t+5)[/latex]
  2. [latex]C(t)=2t(t−3)(t+1)^2[/latex]
  3. [latex]C(t)=4t^4+12t^3−40t^2[/latex]
  4. [latex]f(x)=x^3+x^2−20x[/latex]
  5. [latex]f(x)=x^3+x^2−4x-4[/latex]
  6. [latex]f(x)=2x^3-x^2−8x+4[/latex]
  7. [latex]f(x)=2x^4+6x^2-8[/latex]
  8. [latex]f(x)=x^6-2x^4-3x^2[/latex]
  9. [latex]f(x)=x^5-5x^3+4x[/latex]

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

  1. [latex]f(x)=x^3−9x[/latex], between [latex]x=2[/latex] and [latex]x=4[/latex].
  2. [latex]f(x)=−x^4+4[/latex], between [latex]x=1[/latex] and [latex]x=3[/latex].
  3. [latex]f(x)=x^3−100x+2[/latex], between [latex]x=0.01[/latex] and [latex]x=0.1[/latex].

For the following exercises, find the zeros and give the multiplicity of each.

  1. [latex]f(x)=x^2(2x+3)^5(x−4)^2[/latex]
  2. [latex]f(x)=x^2(x^2+4x+4)[/latex]
  3. [latex]f(x)=(3x+2)^5(x^2−10x+25)[/latex]
  4. [latex]f(x)=x^6-x^5-2x^4[/latex]
  5. [latex]f(x)=4x^5-12x^4+9x^3[/latex]
  6. [latex]f(x)=4x^4(9x^4-12x^3+4x^2)[/latex]

For the following exercises, graph the polynomial functions. Note [latex]x[/latex]– and [latex]y[/latex]– intercepts, multiplicity, and end behavior.

  1. [latex]g(x)=(x+4)(x−1)^2[/latex]
  2. [latex]k(x)=(x−3)^3(x−2)^2[/latex]
  3. [latex]n(x)=−3x(x+2)(x−4)[/latex]

For the following exercises, use the graphs to write the formula for a polynomial function of least degree.

  1. Graph of a negative odd-degree polynomial with zeros at x=-3, 1, and 3.
  2. Graph of a positive odd-degree polynomial with zeros at x=-2, and 3.

For the following exercises, use the graph to identify zeros and multiplicity.

  1. Graph of a negative even-degree polynomial with zeros at x=-4, -2, 1, and 3.
  2. Graph of a positive even-degree polynomial with zeros at x=-2,, and 3.

For the following exercises, use the given information about the polynomial graph to write the equation.

  1. Degree [latex]3[/latex]. Zeros at [latex]x=–2, x=1[/latex], and [latex]x=3[/latex]. [latex]y[/latex]-intercept at [latex](0,–4)[/latex].
  2. Degree [latex]5[/latex]. Roots of multiplicity [latex]2[/latex] at [latex]x=3[/latex] and [latex]x=1[/latex], and a root of multiplicity [latex]1[/latex] at [latex]x=–3[/latex]. [latex]y[/latex]-intercept at [latex](0,9)[/latex].
  3. Degree [latex]5[/latex]. Double zero at [latex]x=1[/latex], and triple zero at [latex]x=3[/latex]. Passes through the point [latex](2,15)[/latex].
  4. Degree 3. Zeros at [latex]x=-3[/latex], [latex]x=-2[/latex] and [latex]x=1[/latex]. y-intercept at [latex](0,12)[/latex].
  5. Degree 4. Roots of multiplicity 2 at [latex]x=\dfrac{1}{2}[/latex] and roots of multiplicity 1 at [latex]x=6[/latex] and [latex]x=-2[/latex]. y-intercept at [latex](0,18)[/latex].

Dividing Polynomials

For the following exercises, use long division to divide. Specify the quotient and the remainder.

  1. [latex](x^2+5x−1)÷(x−1)[/latex]
  2. [latex](3x^2+23x+14)÷(x+7)[/latex]
  3. [latex](6x^2−25x−25)÷(6x+5)[/latex]
  4. [latex](2x^2−3x+2)÷(x+2)[/latex]

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)

  1. [latex](2x^3−6x^2−7x+6)÷(x−4)[/latex]
  2. [latex](4x^3−12x^2−5x−1)÷(2x+1)[/latex]
  3. [latex](3x^3−2x^2+x−4)÷(x+3)[/latex]
  4. [latex](2x^3+7x^2−13x−3)÷(2x−3)[/latex]
  5. [latex](4x^3-5x^2+13)÷(x+4)[/latex]
  6. [latex](x^3-21x^2+147x−343)÷(x-7)[/latex]

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

  1. [latex]x−2[/latex], [latex]3x^4−6x^3−5x+10[/latex]
  2. [latex]x−2[/latex], [latex]4x^4−15x^2−4[/latex]
  3. [latex]x+\frac{1}{3}[/latex], [latex]3x^4+x^3−3x+1[/latex]

Zeros of Polynomial Functions

For the following exercises, use the Remainder Theorem to find the remainder.

  1. [latex](3x^3−2x^2+x−4)÷(x+3)[/latex]
  2. [latex](−3x^2+6x+24)÷(x−4)[/latex]
  3. [latex](x^4−1)÷(x−4)[/latex]
  4. [latex](4x^3+5x^2−2x+7)÷(x+2)[/latex]

For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

  1. [latex]f(x)=2x^3+x^2−5x+2; x+2[/latex]
  2. [latex]f(x)=2x^3+3x^2+x+6; x+2[/latex]
  3. [latex]x^3+3x^2+4x+12; x+3[/latex]
  4. [latex]2x^3+5x^2−12x−30; 2x+5[/latex]

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

  1. [latex]2x^3+7x^2−10x−24=0[/latex]
  2. [latex]x^3+5x^2−16x−80=0[/latex]
  3. [latex]2x^3−3x^2−32x−15=0[/latex]
  4. [latex]2x^3−3x^2−x+1=0[/latex]
  5. [latex]2x^3−5x^2+9x-9=0[/latex]

For the following exercises, find all complex solutions (real and non-real).

  1. [latex]x^3-8x^2+25x-26=0[/latex] Solutions: [latex]x=2[/latex], [latex]x=3\pm i[/latex]
  2. [latex]3x^3-4x^2+11x+10=0[/latex] Solutions: [latex]x=-1[/latex], [latex]x=\dfrac{2\pm i\sqrt{11}}{3}[/latex]
  3. [latex]2x^3-3x^2+32x+17=0[/latex] Solutions: [latex]x=-1[/latex], [latex]x=2\pm2i[/latex]

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

  1. [latex]f(x)=x^4-x^2-1[/latex]
  2. [latex]f(x)=x^3-2x^2+x-1[/latex]
  3. [latex]f(x)=2x^3+37x^2+200x+300[/latex]
  4. [latex]f(x)=2x^4-5x^3-5x^2+5x+3[/latex]
  5. [latex]f(x)=10x^4-21x^2+11[/latex]

For the following exercises, list all possible rational zeros for the functions.

  1. [latex]f(x)=2x^3+3x^2−8x+5[/latex]
  2. [latex]f(x)=6x^4−10x^2+13x+1[/latex]

For the following exercises, find the dimensions of the box described.

  1. The length is twice as long as the width. The height is [latex]2[/latex] inches greater than the width. The volume is [latex]192[/latex] cubic inches.
  2. The length is one inch more than the width, which is one inch more than the height. The volume is [latex]86.625[/latex] cubic inches.
  3. The length is [latex]3[/latex] inches more than the width. The width is [latex]2[/latex] inches more than the height. The volume is [latex]120[/latex] cubic inches.

For the following exercises, find the dimensions of the right circular cylinder described.

  1. The height is one less than one half the radius. The volume is [latex]72\pi[/latex] cubic meters.
  2. The radius and height differ by two meters. The height is greater and the volume is [latex]28.125\pi[/latex] cubic meters.