Algebra 2 Polynomial Division Worksheet with Answers – PDF

Algebra 2 Polynomial Division Worksheet: problems 1–5

1. Divide by a monomial

Divide.

1. \((12x^{3}-8x^{2}+20x): 4x\)
2. \((9x^{4}+6x^{3}-15x^{2}): 3x^{2}\)

2. Missing side of a rectangle

The rectangle has area \(x^{2}+7x+12\) and one side of length \(x+3\). Find the length of the other side.

3. Long division, zero remainder

Divide using long division.

1. \((x^{2}-5x-14): (x+2)\)
2. \((x^{2}+11x+28): (x+4)\)

4. Long division with a remainder

Divide. Write each answer in the form quotient \(+\) remainder over divisor.

1. \((x^{2}+3x-2): (x-1)\)
2. \((x^{2}-6x+4): (x-3)\)

5. Cubic by long division

Divide \(x^{3}+4x^{2}-3x-18\) by \(x-2\) using long division.

Algebra 2 Polynomial Division Worksheet: problems 6–10

6. Division with missing terms

Rewrite each dividend with zero coefficients for the missing powers, then divide.

1. \((x^{3}-8): (x-2)\)
2. \((x^{3}+27): (x+3)\)

7. Fill in a synthetic division table

The table sets up synthetic division of \(x^{3}-6x^{2}+11x-6\) by \(x-1\). Complete the table and write the quotient and the remainder.

8. Synthetic division with a remainder

Use synthetic division to divide \(2x^{3}-5x^{2}+x+7\) by \(x-3\).

9. Divide by x plus a number

Use synthetic division. Remember that \(x+2\) means \(c=-2\).

1. \((x^{3}+5x^{2}-2x-24): (x+2)\)
2. \((x^{4}-3x^{2}+x+5): (x+1)\)

10. Remainder Theorem

Let \(p(x)=x^{3}-4x^{2}+2x+5\). Use the Remainder Theorem to find the remainder when \(p(x)\) is divided by each divisor. Do not divide.

1. \(x-3\)
2. \(x+1\)
3. \(x-2\)

Algebra 2 Polynomial Division Worksheet: problems 11–15

11. Use the Factor Theorem

Let \(f(x)=x^{3}-3x^{2}-10x+24\). Decide whether each expression is a factor of \(f(x)\). Justify each answer.

1. \(x-2\)
2. \(x-3\)
3. \(x+3\)

12. Find the missing coefficient

Find the value of \(k\) so that \(x-2\) is a factor of \(x^{3}+kx^{2}-4x-12\).

13. Divide by a binomial with leading coefficient 2

Divide using long division.

1. \((6x^{2}+x-12): (2x+3)\)
2. \((4x^{2}-4x+5): (2x-1)\)

14. Write the division statement

Divide \(p(x)=3x^{3}+2x^{2}-x+4\) by \(x+1\). Then write \(p(x)\) in the form \(d(x)q(x)+r\).

15. Factor a cubic from its graph

The graph of \(f(x)=x^{3}-2x^{2}-5x+6\) shows that \(x=1\) is a zero. Use synthetic division to factor \(f(x)\) completely and list all zeros.

Algebra 2 Polynomial Division Worksheet: problems 16–20

16. Find the remaining zeros

Given that \(x=-1\) and \(x=2\) are zeros of \(g(x)=x^{4}-2x^{3}-7x^{2}+8x+12\), find the other two zeros.

17. Verify a polynomial identity

Show that \((x^{2}+y^{2})^{2}=(x^{2}-y^{2})^{2}+(2xy)^{2}\) for all \(x\) and \(y\) by expanding both sides.

18. Pythagorean triples from the identity

The triangle has legs \(2xy\) and \(x^{2}-y^{2}\) and hypotenuse \(x^{2}+y^{2}\). Let \(x=3\) and \(y=2\).

1. Find the three side lengths.
2. Show that they satisfy the Pythagorean Theorem.
3. Find the three side lengths when \(x=5\) and \(y=2\).

19. Height of a storage box

The volume of a storage box is \(V(x)=x^{3}+9x^{2}+26x+24\) cubic feet. The length is \(x+4\) feet and the width is \(x+2\) feet.

1. Find an expression for the height.
2. Find the dimensions and the volume when \(x=3\).

20. Break-even points

Mei models the profit from printing T-shirts for the school store with \(P(x)=x^{3}-10x^{2}+31x-30\), in hundreds of dollars, where \(x\) is the number of shirts in hundreds. The store breaks even when \(P(x)=0\), and \(x=2\) is one break-even point. Find the other break-even points.

Algebra 2 Polynomial Division Worksheet: problems 21–22

21. Find two unknown coefficients

When \(p(x)=x^{3}+ax^{2}+bx-6\) is divided by \(x-1\), the remainder is \(0\). When it is divided by \(x+1\), the remainder is \(-10\). Find \(a\) and \(b\).

22. Garden dimensions

The area of Aisha’s rectangular garden is \(A(x)=2x^{3}+7x^{2}-5x-4\) square feet, and its width is \(2x+1\) feet.

1. Find an expression for the length.
2. Factor the length completely.
3. Find the length, width, and area when \(x=3\).

How do the long division problems progress?

Problem 1 divides by a monomial, so each term splits on its own. Next, a rectangle with area \(x^{2}+7x+12\) and one side \(x+3\) asks for the missing side. Problem 6 hides zero coefficients in \(x^{3}-8\). Write them in before you start. Problem 13 raises the stakes with divisors such as \(2x+3\), where the leading coefficient is not 1.

Where do students slip with synthetic division?

The classic slip is the sign of \(c\). In problem 9, \(x+2\) means \(c=-2\), not 2. Another slip is skipping a missing power in a quartic. Likewise, problem 7 gives an incomplete table for \(x^{3}-6x^{2}+11x-6\), so fill each row and read the quotient from the bottom. A final zero means the divisor is a factor.

What do the theorem and application problems add?

Problem 12 finds \(k\) so that \(x-2\) is a factor. Problem 16 gives two zeros of a quartic, so divide twice. Meanwhile, problem 18 sets \(x=3\) and \(y=2\) in the identity, which produces a 5-12-13 triangle. Finally, the garden and storage box problems turn volume and area into missing dimensions.

How do you finish strong?

Multiply back at least three quotients to confirm they reproduce the dividend. Afterward, mark problems where your sign of \(c\) was wrong. Then use the answer key and move on to the unit test.

See every chapter on the Algebra 2 math page.