Polynomial and Rational Functions in Depth: Precalculus Math Worksheet – PDF Download

Precalculus Polynomial Rational Worksheet: problems 1–5

Problem 1: List possible rational zeros

List every possible rational zero of \(P(x)=3x^{4}+2x^{3}-5x+8\).

Problem 2: Rational zeros of a cubic

Find all zeros of \(P(x)=x^{3}-4x^{2}+x+6\).

Problem 3: Factor with the Rational Root Theorem

The graph shows \(P(x)=4x^{3}-12x^{2}-x+3\), which crosses the x-axis three times .

1. List the possible rational zeros.
2. Find all zeros and write \(P(x)\) in factored form.

Problem 4: Counting zeros

Let \(f(x)=(x-1)^{2}(x+4)^{3}(x^{2}+1)\).

1. State the degree of \(f\).
2. List every complex zero with its multiplicity.
3. How many distinct real zeros does \(f\) have?

Problem 5: Complex zeros of a cubic

Find all real and complex zeros of \(P(x)=x^{3}-2x^{2}+4x-8\).

Precalculus Polynomial Rational Worksheet: problems 6–10

Problem 6: Build a polynomial from zeros

Write a monic cubic polynomial with real coefficients whose zeros include \(3\) and \(2-i\). Write it in standard form.

Problem 7: Use a given complex zero

One zero of \(P(x)=x^{4}-2x^{3}+6x^{2}-2x+5\) is \(1+2i\). Find the other three zeros and plot the nonreal zeros on a complex plane.

Problem 8: End behavior of polynomials

Describe the end behavior of each function by completing: as \(x\to-\infty\), \(f(x)\to\ ?\) and as \(x\to\infty\), \(f(x)\to\ ?\).

1. \(f(x)=-2x^{5}+3x^{2}-1\)
2. \(f(x)=x^{4}-8x^{2}+3\)
3. \(f(x)=\frac{1}{2}x^{3}-7x\)

Problem 9: Zeros and multiplicity from a graph

The graph shows a polynomial \(h\) with zeros at \(-3\), \(1\) and \(2\).

1. State the multiplicity of each zero as odd or even and explain how you know.
2. State the least possible degree.
3. Is the leading coefficient positive or negative?

Problem 10: Asymptotes of a rational function

Find all vertical and horizontal asymptotes of \(f(x)=\frac{3x^{2}+1}{x^{2}-4}\).

Precalculus Polynomial Rational Worksheet: problems 11–15

Problem 11: Slant asymptote by division

Find the slant asymptote of \(f(x)=\frac{x^{2}+3x-1}{x-1}\).

Problem 12: Slant asymptote with a negative remainder term

Find the slant asymptote and the vertical asymptote of \(f(x)=\frac{2x^{2}-5x+3}{x+2}\).

Problem 13: Analyze a rational graph

The graph shows \(f(x)=\frac{x^{2}-4}{x-1}\).

1. Write the domain in interval notation.
2. Find the equations of the vertical and slant asymptotes.
3. Find the x-intercepts and the y-intercept.

Problem 14: Quadratic inequality

Solve \(x^{2}-x-6 > 0\) and write the answer in interval notation.

Problem 15: Cubic inequality

Solve \((x+1)(x-2)(x-4) \le 0\).

Precalculus Polynomial Rational Worksheet: problems 16–20

Problem 16: Inequality with a repeated factor

Solve \((x-2)^{2}(x+1) \le 0\).

Problem 17: Rational inequality with zero on one side

Solve \(\frac{x-1}{x+3} \ge 0\).

Problem 18: Rational inequality with a constant on the right

Solve \(\frac{2x}{x-1} < 3\).

Problem 19: Gift box volume

Priya designs a gift box with height \(x\) feet, width \(x+3\) feet and length \(x+4\) feet. The volume must be 60 cubic feet.

1. Write an equation of the form \(P(x)=0\).
2. Use the Rational Root Theorem to find the real solution.
3. Find the other two zeros and the box dimensions.

Problem 20: Average cost

A bakery’s average cost per cookie box is \(A(x)=\frac{20x+1500}{x}\) dollars when it makes \(x\) boxes.

1. Find the horizontal asymptote and explain what it means.
2. Find \(A(100)\).
3. How many boxes must be made so that the average cost is at most $25?

Precalculus Polynomial Rational Worksheet: problems 21–24

Problem 21: Profit model

The profit of a school store is \(P(x)=-x^{3}+14x^{2}-40x\) thousand dollars when \(x\) hundred items are sold, for \(0 < x < 11\).

1. Factor \(P(x)\) completely.
2. Solve \(P(x) > 0\) for \(0 < x < 11\).
3. Find the profit when 600 items are sold.

Problem 22: Polynomial with an imaginary zero

Write a monic polynomial of degree 4 with real coefficients that has zeros \(2\), \(-1\) and \(3i\). Write it in standard form and describe its end behavior.

Problem 23: Rational inequality with a quadratic

Solve \(\frac{x^{2}-x-6}{x-1} \ge 0\).

Problem 24: Roller coaster height

The height of a roller coaster track above ground level, in tens of feet, is \(h(x)=x^{3}-6x^{2}+11x-6\) at horizontal distance \(x\) hundred feet, for \(0 \le x \le 4\). Find where the track is below ground level.

How to use this precalculus polynomial rational worksheet

Print this precalculus polynomial rational worksheet as a PDF, solve every problem on paper, then compare with the answer key.

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