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Operations with Rational Expressions

Adding, subtracting, multiplying, and dividing rationals

Written and reviewed by the Study Mondo Education TeamLast updated
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Operations with Rational Expressions

Multiplying Rational Expressions

ab⋅cd=a⋅cb⋅d\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

Steps:

  1. Factor everything
  2. Multiply numerators and denominators
  3. Cancel common factors
  4. Simplify

Dividing Rational Expressions

ab÷cd=ab⋅dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

Multiply by the reciprocal!

Adding/Subtracting (Same Denominator)

ac±bc=a±bc\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

Combine numerators, keep denominator.

Adding/Subtracting (Different Denominators)

  1. Find the LCD (Least Common Denominator)
  2. Rewrite each fraction with the LCD
  3. Add or subtract numerators
  4. Simplify

Example: 2x+3x+1\frac{2}{x} + \frac{3}{x + 1}

LCD = x(x+1)x(x + 1)

2(x+1)x(x+1)+3xx(x+1)=2x+2+3xx(x+1)=5x+2x(x+1)\frac{2(x + 1)}{x(x + 1)} + \frac{3x}{x(x + 1)} = \frac{2x + 2 + 3x}{x(x + 1)} = \frac{5x + 2}{x(x + 1)}

📚 Practice Problems

1Problem 1easy

❓ Question:

Multiply: x+2x−3⋅x−3x+5\frac{x + 2}{x - 3} \cdot \frac{x - 3}{x + 5}

💡 Show Solution

Multiply numerators and denominators: (x+2)(x−3)(x−3)(x+5)\frac{(x + 2)(x - 3)}{(x - 3)(x + 5)}

Cancel the common factor (x−3)(x - 3): =x+2x+5= \frac{x + 2}{x + 5}

Answer: x+2x+5\frac{x + 2}{x + 5}

2Problem 2easy

❓ Question:

Multiply: x+2x−3⋅x−3x+5\frac{x + 2}{x - 3} \cdot \frac{x - 3}{x + 5}

💡 Show Solution

Multiply numerators and denominators: (x+2)(x−3)(x−3)(x+5)\frac{(x + 2)(x - 3)}{(x - 3)(x + 5)}

Cancel the common factor (x−3)(x - 3): =x+2x+5= \frac{x + 2}{x + 5}

Answer: x+2x+5\frac{x + 2}{x + 5}

3Problem 3medium

❓ Question:

Divide: x2−4x+1÷x+2x2−1\frac{x^2 - 4}{x + 1} \div \frac{x + 2}{x^2 - 1}

💡 Show Solution

Step 1: Multiply by the reciprocal x2−4x+1⋅x2−1x+2\frac{x^2 - 4}{x + 1} \cdot \frac{x^2 - 1}{x + 2}

Step 2: Factor everything (x+2)(x−2)x+1⋅(x+1)(x−1)x+2\frac{(x + 2)(x - 2)}{x + 1} \cdot \frac{(x + 1)(x - 1)}{x + 2}

Step 3: Cancel (x+2)(x + 2) and (x+1)(x + 1) =(x−2)(x−1)1= \frac{(x - 2)(x - 1)}{1}

Step 4: Multiply =(x−2)(x−1)=x2−3x+2= (x - 2)(x - 1) = x^2 - 3x + 2

Answer: x2−3x+2x^2 - 3x + 2

4Problem 4medium

❓ Question:

Divide: x2−4x+1÷x+2x2−1\frac{x^2 - 4}{x + 1} \div \frac{x + 2}{x^2 - 1}

💡 Show Solution

Step 1: Multiply by the reciprocal x2−4x+1⋅x2−1x+2\frac{x^2 - 4}{x + 1} \cdot \frac{x^2 - 1}{x + 2}

Step 2: Factor everything (x+2)(x−2)x+1⋅(x+1)(x−1)x+2\frac{(x + 2)(x - 2)}{x + 1} \cdot \frac{(x + 1)(x - 1)}{x + 2}

Step 3: Cancel (x+2)(x + 2) and (x+1)(x + 1) =(x−2)(x−1)1= \frac{(x - 2)(x - 1)}{1}

Step 4: Multiply =(x−2)(x−1)=x2−3x+2= (x - 2)(x - 1) = x^2 - 3x + 2

Answer: x2−3x+2x^2 - 3x + 2

5Problem 5hard

❓ Question:

Add: 3x−2+4x+1\frac{3}{x - 2} + \frac{4}{x + 1}

💡 Show Solution

Step 1: Find LCD LCD=(x−2)(x+1)\text{LCD} = (x - 2)(x + 1)

Step 2: Rewrite with LCD 3(x+1)(x−2)(x+1)+4(x−2)(x−2)(x+1)\frac{3(x + 1)}{(x - 2)(x + 1)} + \frac{4(x - 2)}{(x - 2)(x + 1)}

Step 3: Add numerators =3(x+1)+4(x−2)(x−2)(x+1)= \frac{3(x + 1) + 4(x - 2)}{(x - 2)(x + 1)}

Step 4: Expand and simplify =3x+3+4x−8(x−2)(x+1)= \frac{3x + 3 + 4x - 8}{(x - 2)(x + 1)} =7x−5(x−2)(x+1)= \frac{7x - 5}{(x - 2)(x + 1)}

Answer: 7x−5(x−2)(x+1)\frac{7x - 5}{(x - 2)(x + 1)}

6Problem 6hard

❓ Question:

Add: 3x−2+4x+1\frac{3}{x - 2} + \frac{4}{x + 1}

💡 Show Solution

Step 1: Find LCD LCD=(x−2)(x+1)\text{LCD} = (x - 2)(x + 1)

Step 2: Rewrite with LCD 3(x+1)(x−2)(x+1)+4(x−2)(x−2)(x+1)\frac{3(x + 1)}{(x - 2)(x + 1)} + \frac{4(x - 2)}{(x - 2)(x + 1)}

Step 3: Add numerators =3(x+1)+4(x−2)(x−2)(x+1)= \frac{3(x + 1) + 4(x - 2)}{(x - 2)(x + 1)}

Step 4: Expand and simplify =3x+3+4x−8(x−2)(x+1)= \frac{3x + 3 + 4x - 8}{(x - 2)(x + 1)} =7x−5(x−2)(x+1)= \frac{7x - 5}{(x - 2)(x + 1)}

Answer: 7x−5(x−2)(x+1)\frac{7x - 5}{(x - 2)(x + 1)}

Explain using:

⚠️ Common Mistakes: Operations with Rational Expressions

Avoid these 3 frequent errors

🌍 Real-World Applications: Operations with Rational Expressions

See how this math is used in the real world

📝 Worked Example: Solving a Quadratic by Factoring

Problem:

Solve x2−5x+6=0x^2 - 5x + 6 = 0.

2Factor the quadratic
3Set each factor equal to zero
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Solving Rational Equations
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📌 Related Topics in Rational Expressions

Simplifying Rational ExpressionsSolving Rational Equations

❓ Frequently Asked Questions

What is Operations with Rational Expressions?▾
Adding, subtracting, multiplying, and dividing rationals
How can I study Operations with Rational Expressions effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 6 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Operations with Rational Expressions study guide free?▾
Yes — all study notes, flashcards, and practice problems for Operations with Rational Expressions on Study Mondo are free to access. No account is needed.
What course covers Operations with Rational Expressions?▾
Operations with Rational Expressions is part of the Algebra 2 course on Study Mondo, specifically in the Rational Expressions section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Operations with Rational Expressions?▾
Yes, this page includes 6 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.