Operations with Rational Expressions

Adding, subtracting, multiplying, and dividing rationals

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 2medium

❓ 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

3Problem 3hard

❓ 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)}

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