← Back to the Polynomial Division test
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Each solution shows the division or substitution and the scoring guide. Accept any correct method, such as the Remainder Theorem in place of division when only a remainder is requested, and award partial credit for correct setup.
Standards: A-APR.B.2, A-APR.D.6, A-APR.C.4 · Level: Grade 11 Algebra 2
Question 1 Long division (3 points)
Divide x3 by x to get x2. Multiply: x2(x + 3) = x3 + 3x2. Subtract: −5x2 − 9x + 25.
Divide −5x2 by x to get −5x. Multiply: −5x(x + 3) = −5x2 − 15x. Subtract: 6x + 25.
Divide 6x by x to get 6. Multiply: 6(x + 3) = 6x + 18. Subtract: 7.
Quotient x2 − 5x + 6, remainder 7
Scoring: 1 point for each correct division step, up to 2, and 1 point for the quotient and remainder.
Question 2 Synthetic division (3 points)
The x3 term is missing, so use the coefficients 3, 0, −2, 5, −9.
Use r = −2. Bring down 3. Then multiply by −2 and add down each column.
Bottom row: 3, −6, 10, −15, 21.
Quotient 3x3 − 6x2 + 10x − 15, remainder 21
Scoring: 1 point for including the zero coefficient, 1 for the table, 1 for the quotient and remainder.
Question 3 Remainder Theorem (2 points)
f(2) = 16 − 24 + 4 − 5 = −9.
f(−1) = 1 + 3 − 2 − 5 = −3.
Remainders −9 and −3
Scoring: 1 point each remainder.
Question 4 Factor Theorem (3 points)
f(4) = 64 − 80 − 8 + 24 = 0, so x − 4 is a factor.
Use r = 4. Bring down 1. Then multiply by 4 and add down each column.
Bottom row: 1, −1, −6, 0.
Quotient x2 − x − 6, remainder 0
x2 − x − 6 = (x − 3)(x + 2).
f(x) = (x − 4)(x − 3)(x + 2)
Scoring: 1 point for f(4) = 0, 1 for the quotient, 1 for complete factoring.
Question 5 An unknown coefficient (2 points)
The value at x = −3 must be 0: −27 + 9k + 6 + 12 = 0, so 9k − 9 = 0.
k = 1
Scoring: 1 point for the equation, 1 point for k.
Question 6 Polynomial identity (3 points)
a) Left: x4 + 2x2y2 + y4. Right: x4 − 2x2y2 + y4 + 4x2y2 = x4 + 2x2y2 + y4. Equal.
b) Legs 16 − 1 = 15 and 2(4)(1) = 8; hypotenuse 16 + 1 = 17. Check: 225 + 64 = 289.
Sides 15, 8, 17
Scoring: 1 point for each expansion in (a), 1 point for (b).
Question 7 Remainders from a graph (2 points)
a) By the Remainder Theorem the remainders are f(2) = −3 and f(3) = 0.
b) The zeros −1, 1, and 3 give the factors x + 1, x − 1, x − 3.
Scoring: 1 point each part.
Question 8 Volume of a prism (2 points)
Divide x3 by x to get x2. Multiply: x2(x + 2) = x3 + 2x2. Subtract: 7x2 + 26x + 24.
Divide 7x2 by x to get 7x. Multiply: 7x(x + 2) = 7x2 + 14x. Subtract: 12x + 24.
Divide 12x by x to get 12. Multiply: 12(x + 2) = 12x + 24. Subtract: 0.
Quotient x2 + 7x + 12, remainder 0
x2 + 7x + 12 = (x + 3)(x + 4).
Base area x2 + 7x + 12; length x + 4, width x + 3
Scoring: 1 point for the division, 1 point for the factoring.