Polynomial Division Test – Algebra 2 Math (Printable PDF)

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.

1. Prove (x² + y²)² = (x² − y²)² + (2xy)² by expanding. (2 points)
2. 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).

1. State the remainder when f is divided by x − 2 and by x − 3. (1 point)
2. 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.