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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

Practice problems

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

Show solution
  1. Polynomial Operations: First identify exactly what the question is asking: Simplify (3x^2 + x) + (4x^2 - 6x).
  2. For polynomial work, use degree, leading terms, and like terms to keep the expression organized.
  3. Combine like terms.
  4. Check the result by substituting or estimating: the response should match 7x^2 - 5x and make sense in the original problem.

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. Polynomial Operations: First identify exactly what the question is asking: Multiply (x + 2)(x - 9).
  2. For polynomial work, use degree, leading terms, and like terms to keep the expression organized.
  3. The middle terms are -9x and 2x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x^2 - 7x - 18

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

Show solution
  1. Division: First identify exactly what the question is asking: Divide (12x^3 - 6x^2) by 3x^2.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Divide each term.
  4. Check the result by substituting or estimating: the response should match 4x - 2 and make sense in the original problem.

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. Factoring: First identify exactly what the question is asking: Factor x^3 - 16x.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Factor x, then difference of squares.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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. Rational Expressions: First identify exactly what the question is asking: Simplify (x^2 - 36)/(x - 6).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Factor difference of squares.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x + 6, x != 6

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