Polynomial Division and the Remainder Theorem: Algebra 2 Math Worksheet – PDF Download

Problem 1: Divide by a monomial

Divide.

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

Problem 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.

Problem 3: Long division, zero remainder

Divide using long division.

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

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

Problem 5: Cubic by long division

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

Problem 6: Division with missing terms

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

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

Problem 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.

Problem 8: Synthetic division with a remainder

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

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

Problem 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. \(x-2\)

Problem 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. \(x+3\)

Problem 12: Find the missing coefficient

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

Problem 13: Divide by a binomial with leading coefficient 2

Divide using long division.

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

Problem 14: Write the division statement

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

Problem 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.

Problem 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.

Problem 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.

Problem 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 \(y=2\).

Problem 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 \(x=3\).

Problem 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.

Problem 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 \(b\).

Problem 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 \(x=3\).

See every chapter on the Algebra 2 math page.