Precalculus Polynomial Division and the Factor Theorem Worksheet (PDF)

Precalculus math worksheet: Polynomial Division and the Factor Theorem – Maths PDF

Cover of the book Precalculus Math Worksheets with Answers (PDF)

Download as a PDFThe Precalculus worksheets book with answersPrintable PDFSee the book ›

Sommaire

This precalculus polynomial division worksheet for Precalculus covers polynomial division and the factor theorem (Common Core A-APR.B.2, A-APR.D.6, A-APR.B.3), with a free PDF to print.

These 20 problems practice dividing polynomials and using the Factor Theorem (Common Core A-APR.B.2, A-APR.D.6 and A-APR.B.3). You will use long and synthetic division, the Remainder Theorem and the Rational Root Theorem.

Precalculus · Common Core State Standards: A-APR.B.2, A-APR.D.6, A-APR.B.3

Precalculus polynomial division worksheet: exercises 1 to 10

Exercise 1 – Divide a quadratic

Divide \(x^{2}+7x+12\) by \(x+3\).

Exercise 2 – Long division, leading coefficient 2

Divide \(2x^{2}+5x-3\) by \(x+3\).

Exercise 3 – Remainder from long division

Divide \(x^{3}+2x^{2}-4x+7\) by \(x-1\). Write the answer as quotient plus remainder over divisor.

Exercise 4 – Synthetic division

Use synthetic division to divide \(x^{3}-4x^{2}+x+6\) by \(x-2\).

Exercise 5 – Synthetic division with c = -1

Divide \(2x^{3}-3x^{2}+5x-1\) by \(x+1\) with synthetic division.

Exercise 6 – Remainder Theorem

Find the remainder when \(f(x)=x^{4}-3x^{2}+5\) is divided by \(x-2\), without dividing.

Exercise 7 – Is it a factor?

Graph of x^3 – 27.

Is \(x-3\) a factor of \(x^{3}-27\)? If so, factor completely over the integers.

Exercise 8 – Test a factor

Is \(x+2\) a factor of \(x^{3}+3x^{2}-4x-12\)?

Exercise 9 – Find an unknown coefficient

Find \(k\) so that \(x-2\) is a factor of \(x^{3}+kx^{2}-4x-4\).

Exercise 10 – Factor completely

Given that \(x=1\) is a zero, factor \(x^{3}-7x+6\) completely.

Exercises 11 and up

Exercise 11 – Rational root candidates

List all possible rational zeros of \(2x^{3}-3x^{2}-11x+6\).

Exercise 12 – Find all zeros

Graph of f.

Find all zeros of \(f(x)=2x^{3}-3x^{2}-11x+6\).

Exercise 13 – Missing terms

Divide \(x^{3}-8\) by \(x-2\) using synthetic division. Remember the missing terms.

Exercise 14 – Biquadratic

Solve \(x^{4}-5x^{2}+4=0\).

Exercise 15 – Quartic zeros

Find all zeros of \(x^{4}-x^{3}-7x^{2}+x+6\), given that \(1\) and \(-1\) are zeros.

Exercise 16 – Divide by a non-monic linear divisor

Divide \(6x^{2}+x-5\) by \(2x+3\).

Exercise 17 – Box dimensions

Volume V = x^3 + 9x^2 + 26x + 24.

A storage box has volume \(V(x)=x^{3}+9x^{2}+26x+24\) cubic feet and height \(x+2\) feet. Find expressions for the other two dimensions.

Exercise 18 – Break-even years

Profit model for Liam’s food truck.

The profit of Liam’s food truck, in thousands of dollars, is \(P(t)=t^{3}-5t^{2}-2t+24\) after \(t\) years. Show that \(P(4)=0\), then find every \(t\ge 0\) at which the profit is \(0\).

Exercise 19 – Multiplicity of a zero

Graph of f.

Show that \(x=\frac12\) is a zero of \(4x^{3}-8x^{2}+5x-1\), then find all zeros and the multiplicity of each.

Exercise 20 – Polynomial with integer coefficients

Write a cubic with integer coefficients that has zeros \(2\), \(-\frac12\) and \(3\) and leading coefficient \(2\).

Key facts behind this precalculus polynomial division worksheet

Key facts

f(x) = d(x)q(x) + r(x), with r of lower degree than d.
Remainder Theorem: dividing by x – c leaves f(c).
Factor Theorem: x – c is a factor if and only if f(c) = 0.
Rational Root Theorem: p divides the constant term and q divides the leading coefficient.
Use 0 as a placeholder coefficient for each missing power.

How to use this precalculus polynomial division worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the precalculus polynomial division worksheet as a PDF with the button below and keep it with your notes.

All Precalculus chapters: Precalculus math.

Voter.. post

Free printable PDF