15.3 Operations with algebraic fractions — Learn

Prerequisites: simplify fractions, factorise expressions, and find common denominators.

Objectives: add, subtract, multiply and divide algebraic fractions, then simplify and state restrictions.

For addition and subtraction, use a common denominator. For multiplication, multiply numerators and denominators. For division, multiply by the reciprocal, which is the fraction turned upside down.

Example 1. Simplify $\frac{2}{x}+\frac{3}{x+1}$.

Use common denominator $x(x+1)$. Rewrite the fractions: $\frac{2(x+1)}{x(x+1)}+\frac{3x}{x(x+1)}$. Add the numerators: $\frac{2x+2+3x}{x(x+1)}$. Collect terms: $\frac{5x+2}{x(x+1)}$. The original denominators require $x\neq0,-1$.

Example 2. Simplify $\frac{x}{3}\div\frac{2x}{5}$.

Multiply by the reciprocal: $\frac{x}{3}\times\frac{5}{2x}$. Cancel the common factor $x$: $\frac{5}{6}$. Here $x\neq0$, since the divisor must not be zero.

Watch out. Adding denominators is not a valid fraction method. Use a common denominator and combine only the numerators.

Example 3. Simplify $\frac{5}{x-1}-\frac{2}{x+1}$.

Use the common denominator $(x-1)(x+1)$. Rewrite the fractions: $\frac{5(x+1)}{(x-1)(x+1)}-\frac{2(x-1)}{(x-1)(x+1)}$. Subtract the numerators: $\frac{5x+5-2x+2}{(x-1)(x+1)}$. Collect terms: $\frac{3x+7}{(x-1)(x+1)}$, where $x\neq1,-1$.

Example 4. Simplify $\frac{2x}{3}\times\frac{9}{4x}$.

Multiply the numerators and denominators: $\frac{18x}{12x}$. Cancel the common factor $6x$: $\frac{3}{2}$, where $x\neq0$.

Progress check

Before simplifying a quotient, identify values that make either original denominator zero or make the divisor zero.

You should now be able to

• add or subtract algebraic fractions using a common denominator;
• multiply algebraic fractions and divide by multiplying by the reciprocal;
• simplify results and state excluded values.