Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
These problems cover long division and synthetic division of polynomials, the Remainder and Factor Theorems, and polynomial identities. They address Common Core standards A-APR.B.2, A-APR.D.6, and A-APR.C.4.
Early problems divide step by step; later ones use missing terms, non-monic divisors, graphs, and modeling contexts. Check every quotient by multiplying it by the divisor and adding the remainder.
Standards: A-APR.B.2, A-APR.D.6, A-APR.C.4 · Level: Grade 11 Algebra 2
Problem 1 Long division with no remainder ★★★
Use long division to divide x2 + 11x + 28 by x + 4.
Problem 2 Long division with a leading coefficient ★★★
Divide 2x2 + 5x − 12 by x + 4 using long division.
Problem 3 A remainder appears ★★★
Divide x2 + 6x − 5 by x − 2 and write the result as quotient plus remainder over divisor.
Problem 4 Completing a synthetic division ★★★
The table shows the start of dividing 2x3 − 5x2 − x + 9 by x − 2. Complete the table, then write the quotient and the remainder.
Problem 5 Synthetic division with a zero remainder ★★★
Use synthetic division to divide x3 + 2x2 − 13x + 10 by x − 2.
Problem 6 Synthetic division by x plus a number ★★★
Divide 3x3 − x2 + 4x − 7 by x + 2 using synthetic division. Rewrite the divisor as x − (−2) first.
Problem 7 A missing term ★★★
Divide x4 − 5x2 + 2x + 8 by x − 3 using synthetic division. Write the quotient and remainder.
Problem 8 Dividing by 2x + 1 ★★★
Use long division to divide 6x3 + x2 + 7x + 9 by 2x + 1.
Problem 9 Reading a remainder from a graph ★★★
The graph of f(x) = x3 − 4x2 + x + 6 is shown. Use the Remainder Theorem and the graph to state the remainder when f(x) is divided by x − 4 and by x − 1. Then confirm the first remainder with synthetic division.
Problem 10 Evaluating with the Remainder Theorem ★★★
For f(x) = 2x3 − 5x2 + x − 8, find the remainder when f is divided by x − 3 and by x + 2, without dividing.
Problem 11 Using the Factor Theorem ★★★
Which of x + 3, x − 1, and x + 4 are factors of x3 + x2 − 10x + 8? Show the evaluations.
Problem 12 Finding an unknown coefficient ★★★
For what value of k does x − 2 divide x3 − kx2 + 9x − 14 with no remainder?
Problem 13 Factoring completely from a zero ★★★
The graph of f(x) = x3 − x2 − 10x − 8 is shown and −2 is a zero. Use synthetic division to factor f completely, and confirm the zeros on the graph.
Problem 14 Dividing by a quadratic ★★★
Use long division to divide x4 + 3x3 − 2x2 − 3x + 1 by x2 + 2x − 1.
Problem 15 A divisor with leading coefficient 2 ★★★
Divide 6x3 − 7x2 + 5x − 7 by 2x − 3 using synthetic division with r = 32. Then convert the result to a quotient and remainder for the divisor 2x − 3.
Problem 16 Dimensions of a box ★★★
A rectangular box has volume V(x) = x3 + 8x2 + 19x + 12 cubic inches and height x + 3 inches. Find an expression for the area of the base, then factor it to find the length and width.
Problem 17 Break-even points ★★★
The net profit of a food truck, in thousands of dollars, is modeled by P(x) = x3 − 7x2 + 14x − 8, where x is the number of hundreds of meals sold. Show that x = 1 gives zero profit, and find all break-even points.
Problem 18 Proving an identity ★★★
Prove that (x2 + y2)2 = (x2 − y2)2 + (2xy)2 by expanding both sides. Then use x = 3 and y = 2 to write a Pythagorean triple and draw the matching triangle labels.
Problem 19 Using identities for calculation ★★★
(a) Use x = 5 and y = 2 in the identity from the previous problem to find a Pythagorean triple. (b) Use the identity a2 − b2 = (a − b)(a + b) to compute 1,0032 − 9972 without a calculator.
Problem 20 Finding two unknown coefficients ★★★
The polynomial p(x) = x3 + ax2 + bx + 6 has x − 1 and x + 2 as factors. Find a and b, and then find the third factor.
