Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
This test covers polynomial long division, synthetic division, the Remainder and Factor Theorems, and polynomial identities. Show each division step, include zero coefficients for missing terms, and check answers when time allows. No calculator is needed. The test is worth 20 points.
Standards: A-APR.B.2, A-APR.D.6, A-APR.C.4 · Level: Grade 11 Algebra 2
Question 1 Long division (3 points)
Use long division to divide x3 − 2x2 − 9x + 25 by x + 3. State the quotient and the remainder.
Question 2 Synthetic division (3 points)
Use synthetic division to divide 3x4 − 2x2 + 5x − 9 by x + 2.
Question 3 Remainder Theorem (2 points)
Let f(x) = x4 − 3x3 + 2x − 5. Find the remainder when f is divided by x − 2 and by x + 1, without dividing.
Question 4 Factor Theorem (3 points)
Show that x − 4 is a factor of f(x) = x3 − 5x2 − 2x + 24, and then factor f(x) completely.
Question 5 An unknown coefficient (2 points)
Find k so that x + 3 is a factor of x3 + kx2 − 2x + 12.
Question 6 Polynomial identity (3 points)
The figure shows a right triangle whose sides are given in terms of x and y.
- Prove (x2 + y2)2 = (x2 − y2)2 + (2xy)2 by expanding. (2 points)
- Find the side lengths when x = 4 and y = 1. (1 point)
Question 7 Remainders from a graph (2 points)
The graph of f(x) = x3 − 3x2 − x + 3 is shown with the point (2, −3).
- State the remainder when f is divided by x − 2 and by x − 3. (1 point)
- Name every linear factor of f that the graph shows. (1 point)
Question 8 Volume of a prism (2 points)
A rectangular prism has volume x3 + 9x2 + 26x + 24 and height x + 2. Find the expression for the area of its base and factor it into length and width.
