Rational Expressions

Grade 9 · algebra · 48 practice problems · read aloud

🔊 Listen to this explanation

Rational Expressions: The Fraction of Algebra

A rational expression is a fraction where the numerator and denominator are polynomials. Think of it as a fancy version of the fractions you already know, but with variables like 'x'. We use them to solve complex problems in engineering, physics, and economics.

🔍 Key Steps to Simplify

  1. Factor Everything: Completely factor both the numerator and the denominator.
  2. State Restrictions: Find the values that make the denominator zero. These are excluded values.
  3. Cancel Common Factors: Cancel any factors that are the same in the top and bottom.
  4. Write Final Answer: Write the simplified expression with its restrictions.

📚 Worked Examples

Example 1: Simplify (3x + 6) / (x² + 4x + 4)

  1. Factor: Numerator: 3(x + 2). Denominator: (x + 2)(x + 2).
  2. Restrictions: (x + 2) = 0, so x ≠ -2.
  3. Cancel: One (x + 2) cancels from top and bottom.
  4. Answer: 3 / (x + 2), where x ≠ -2.

Example 2: Simplify (x² - 9) / (x² - 5x + 6)

  1. Factor: Numerator: (x - 3)(x + 3). Denominator: (x - 2)(x - 3).
  2. Restrictions: x ≠ 2 and x ≠ 3.
  3. Cancel: The (x - 3) factors cancel.
  4. Answer: (x + 3) / (x - 2), where x ≠ 2, 3.

⚠️ Common Mistakes

  • Canceling Terms: You can only cancel factors (things being multiplied), not terms being added/subtracted. (x+1) in the numerator and denominator can cancel, but a standalone 'x' cannot.
  • Forgetting Restrictions: Always state the values that make the original denominator zero. This is a crucial part of the answer.
  • Incorrect Factoring: Solid factoring skills are essential. Practice factoring trinomials and difference of squares.

💡 Tips & Tricks

  • Factor First, Always: Before you do anything else, factor completely.
  • The "Big X": Draw a large X through canceled factors in your work to visually track them.
  • Domain Detective: Find restrictions before canceling, using the original denominator.

🎯 How to Practice

Start with simple expressions that require factoring a GCF or a difference of squares. Then, move to trinomials. Create flashcards with problems on the front and steps on the back. Practice explaining the steps to a friend or family member—teaching is the best way to learn!

Practice problems

6 of the 48, worked through step by step — try them before opening the answer.

1 (x² - 9)/(x + 3) = ?

Hint: Look for a difference of squares pattern in the numerator that can be factored

Show the answer

Answer: x - 3

  1. Factor the numerator: x² - 9 = (x + 3)(x - 3)
  2. Rewrite the expression: [(x + 3)(x - 3)]/(x + 3)
  3. Cancel the common factor (x + 3) from numerator and denominator
  4. The simplified expression is x - 3

The answer is x - 3.

2 (x² - 49)/(x + 7) = ?

Hint: Look for a difference of squares pattern in the numerator that can be factored.

Show the answer

Answer: x - 7

  1. Factor the numerator: x² - 49 = (x + 7)(x - 7)
  2. Rewrite the expression: (x + 7)(x - 7)/(x + 7)
  3. Cancel the common factor (x + 7) from numerator and denominator
  4. The simplified expression is x - 7

The answer is x - 7.

3 (x² - 16)/(x + 4) = ?

Hint: Look for a pattern in the numerator that can be factored using a special product formula.

Show the answer

Answer: x - 4

  1. Recognize that the numerator x² - 16 is a difference of squares.
  2. Factor the numerator: x² - 16 = (x + 4)(x - 4)
  3. Rewrite the expression: [(x + 4)(x - 4)]/(x + 4)
  4. Cancel the common factor (x + 4) from numerator and denominator
  5. The simplified expression is x - 4

The answer is x - 4.

4 (x² - 25)/(x + 5) = ?

Hint: Look for a difference of squares pattern in the numerator that can be factored

Show the answer

Answer: x - 5

  1. Factor the numerator: x² - 25 = (x + 5)(x - 5)
  2. Rewrite the expression: (x + 5)(x - 5)/(x + 5)
  3. Cancel the common factor (x + 5) from numerator and denominator
  4. The simplified expression is x - 5

The answer is x - 5.

5 (x² - 64)/(x + 8) = ?

Hint: Look for a difference of squares pattern in the numerator and see if it can be factored

Show the answer

Answer: x - 8

  1. Recognize that x² - 64 is a difference of squares: x² - 8²
  2. Factor the numerator: (x - 8)(x + 8)
  3. Rewrite the expression: (x - 8)(x + 8)/(x + 8)
  4. Cancel the common factor (x + 8) from numerator and denominator
  5. The simplified expression is x - 8

The answer is x - 8.

6 (x² - 36)/(x + 6) = ?

Hint: Look for a difference of squares pattern in the numerator that can be factored.

Show the answer

Answer: x - 6

  1. Factor the numerator x² - 36 as (x + 6)(x - 6)
  2. Rewrite the expression as [(x + 6)(x - 6)]/(x + 6)
  3. Cancel the common factor (x + 6) from numerator and denominator
  4. The simplified expression is x - 6

The answer is x - 6.

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