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Polynomials and Rational Expressions Checkpoint

A free Algebra II lesson from the “Equations, Rational Functions, and Conics” unit, with a worked example and practice problems including step-by-step solutions.

This checkpoint blends polynomial operations, polynomial division, higher-degree factoring, rational expression simplification, and rational equations.

What you'll learn

Why it matters: This checkpoint integrates higher-degree polynomial work with rational expressions and equations. The two topics share factoring, excluded values, and zeros — engineers and statisticians use them together every time a model has both a numerator and a denominator that need careful handling.

Worked example

Problem. Simplify (x^2 - 16)/(x - 4).

  1. Factor x^2 - 16 as (x - 4)(x + 4).
  2. Cancel x - 4.
  3. The original denominator excludes x = 4.

Answer: x + 4, x != 4

Lesson preview: this page shows 5 practice items. The interactive activity contains 25 practice items and 25 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Simplify (3x^2 + x) + (4x^2 - 6x).

Show solution
  1. Combine like terms.

Answer: 7x^2 - 5x

2. Multiply (x + 2)(x - 9).

Choices: x^2 - 7x - 18 · x^2 + 11x - 18 · x^2 - 18 · 2x - 9

Show solution
  1. The middle terms are -9x and 2x.

Answer: x^2 - 7x - 18

3. Divide (12x^3 - 6x^2) by 3x^2.

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  1. Divide each term.

Answer: 4x - 2

4. Factor x^3 - 16x.

Choices: x(x - 4)(x + 4) · x(x - 16) · (x - 4)(x + 4) · x^2 - 16

Show solution
  1. Factor x, then difference of squares.

Answer: x(x - 4)(x + 4)

5. Simplify (x^2 - 36)/(x - 6).

Choices: x + 6, x != 6 · x - 6 · x + 6, x != -6 · x^2 + 6

Show solution
  1. Factor difference of squares.

Answer: x + 6, x != 6

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