Precalculus Polynomial Division and the Factor Theorem Test with Point Values (PDF)

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

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This precalculus polynomial division test 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.

This 20-point test covers long and synthetic division, the Remainder and Factor Theorems, rational zeros and a box model (A-APR.B.2, A-APR.B.3). You have 45 minutes. Show your work.

Time and points

About this test

Grade: Precalculus
Chapter: Polynomial Division and the Factor Theorem
Standards: A-APR.B.2, A-APR.D.6, A-APR.B.3
Time: 45 minutes
Total: 20 points

Precalculus polynomial division test: questions

Question 1 – Divide and factor [2 pts]

Graph of the polynomial.

Divide \(x^{3}-x^{2}-14x+24\) by \(x-2\) and factor the quotient.

Question 2 – Synthetic division with a remainder [2 pts]

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

Question 3 – Remainder Theorem [2 pts]

Find the remainder when \(f(x)=2x^{3}-5x^{2}+x-7\) is divided by \(x-3\), using the Remainder Theorem.

Question 4 – Factor Theorem [2 pts]

Find \(k\) so that \(x+1\) is a factor of \(x^{3}+kx^{2}-2x+8\).

Question 5 – Find all zeros [3 pts]

Graph of f.

Let \(f(x)=2x^{3}+x^{2}-13x+6\). (a) List the possible rational zeros. (b) Show that \(2\) is a zero. (c) Find all zeros.

Question 6 – Division with a remainder expression [2 pts]

Write \(4x^{2}-3x+1\) as \((x-2)q(x)+r\).

Question 7 – Biquadratic zeros [2 pts]

Find all real zeros of \(x^{4}-13x^{2}+36\).

Question 8 – Dimensions of a box [3 pts]

Box with volume V.

Maya’s box has volume \(V(x)=x^{3}+12x^{2}+44x+48\) cubic inches and height \(x+2\) inches. (a) Divide \(V\) by \(x+2\). (b) Factor the quotient to find the other dimensions. (c) Find the volume when \(x=3\).

Question 9 – Build a polynomial [2 pts]

Write a polynomial with integer coefficients, leading coefficient 2, and zeros \(-\frac12\), \(1\) and \(4\).

Key facts behind this precalculus polynomial division test

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 test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All Precalculus chapters: Precalculus math.

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