Algebra 2 Dividing Polynomials and the Remainder Theorem Worksheet (PDF)

Algebra 2 math worksheet: Dividing Polynomials and the Remainder Theorem – Maths PDF

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This algebra 2 polynomial division worksheet for Algebra 2 covers dividing polynomials and the remainder theorem (Common Core A-APR.B.2, A-APR.D.6), with a free PDF to print.

These 20 problems practice polynomial division and the Remainder Theorem (Common Core A-APR.B.2 and A-APR.D.6). The first problems check long division by a binomial, the middle problems add synthetic division, the Remainder Theorem, and the Factor Theorem, and the last six are word problems and multi-step tasks that need dimension problems and writing quotients with a remainder term.

Work in order and show every step. Write each answer in simplest form, and open the answer key only after you have tried the problem yourself.

Algebra 2 · Common Core State Standards: A-APR.B.2, A-APR.D.6

Algebra 2 polynomial division worksheet: exercises 1 to 10

Exercise 1 – Divide, no remainder

Long division cycle.

Divide using long division: \((x^{2} + 7x + 12) \div (x + 3)\).

Exercise 2 – Divide, remainder

Divide: \((x^{2} – 5x + 7) \div (x – 2)\). Write the answer as a quotient plus a remainder over the divisor.

Exercise 3 – Leading coefficient 2

Divide: \((10x^{2} – x – 24) \div (2x + 3)\).

Exercise 4 – Cubic long division

Divide: \((x^{3} + 2x^{2} – 5x – 6) \div (x + 1)\).

Exercise 5 – Synthetic division

Fill in the blank cells.

Use synthetic division to divide \((2x^{3} – 3x^{2} + x – 5) \div (x – 2)\).

Exercise 6 – Missing term

Divide \(x^{3} – 7x + 6\) by \(x – 2\) using synthetic division. Remember the missing power.

Exercise 7 – Divisor x + 3

Divide \(x^{4} + 3x^{3} – x + 4\) by \(x + 3\). Use \(k = -3\).

Exercise 8 – Remainder Theorem, zero

Find the remainder when \(x^{3} – 4x^{2} + x + 6\) is divided by \(x – 3\), without dividing.

Exercise 9 – Remainder Theorem

Find the remainder when \(2x^{3} + x^{2} – 5x + 4\) is divided by \(x + 2\).

Exercise 10 – Factor check

Decide whether \(x – 4\) is a factor of \(x^{3} – 5x^{2} + 2x + 8\).

Exercises 11 and up

Exercise 11 – Find k

Find \(k\) so that \(x + 1\) is a factor of \(x^{3} + kx^{2} – 5x + 3\).

Exercise 12 – Read the graph

Graph of p(x).

The graph of the cubic \(p(x) = x^{3} – 2x^{2} – 5x + 6\) crosses the x-axis at \(-2\), \(1\), and \(3\). Write \(p(x)\) in factored form.

Exercise 13 – Factor completely

Given that \(x – 1\) is a factor of \(x^{3} – 6x^{2} + 11x – 6\), factor the polynomial completely.

Exercise 14 – Rewrite a rational expression

Rewrite \(\frac{x^{2} + 3x + 5}{x + 2}\) in the form \(q(x) + \frac{r}{x + 2}\).

Exercise 15 – Rewrite with a cubic

Rewrite \(\frac{3x^{3} – 2x^{2} + 4}{x – 1}\) as a quotient plus a remainder term.

Exercise 16 – Box dimensions

Box with unknown height.

A shipping box has volume \(V = x^{3} + 9x^{2} + 26x + 24\) cubic inches. Its length is \(x + 2\) and its width is \(x + 3\). Find the height.

Exercise 17 – Rectangle length

A rectangle has area \(x^{3} + 3x^{2} – 4x – 12 \text{ ft}^{2}\) and width \(x + 3\) feet. Find its length.

Exercise 18 – Evaluate with synthetic division

Synthetic division with k = 4 gives P(4).

A population model is \(P(t) = t^{3} – 6t^{2} + 11t + 2\) (in hundreds). Use synthetic division to find \(P(4)\).

Exercise 19 – Difference of cubes

Show that \(x – 5\) is a factor of \(x^{3} – 125\), then factor completely.

Exercise 20 – No division needed

Without dividing, find the remainder when \(x^{10} – 1\) is divided by \(x – 1\) and by \(x + 1\).

Key facts behind this algebra 2 polynomial division worksheet

Key facts

a(x) / b(x) = q(x) + r(x) / b(x) with deg r < deg b.
Synthetic division works for divisors x – k.
Remainder Theorem: p(x) ÷ (x – k) has remainder p(k).
Factor Theorem: x – k is a factor if and only if p(k) = 0.
Include 0 coefficients for missing powers.

How to use this algebra 2 polynomial division worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the algebra 2 polynomial division worksheet as a PDF with the button below and keep it with your notes.

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