Simplify and combine algebraic fractions

Algebraic fraction operations

Record the original denominator restrictions and exclude zeros of any divisor. Factorise before simplifying, use a reciprocal for division and a common denominator for addition or subtraction

1. Record the original restrictions Every original denominator must be nonzero. When dividing, the divisor must also be nonzero. Cancelling a factor does not restore an excluded input
2. Factorise first Write each numerator and denominator as products before looking for common factors \(\displaystyle \frac{6x\left(x+2\right)}{5\left(x-1\right)}\)
3. Cancel whole factors Cancel a complete common factor only where it is nonzero, and retain the original excluded inputs. Separate addends cannot be cancelled \(\displaystyle \frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+5\right)}=\frac{x+3}{x+5},\quad x\ne-2,-5\)
4. Use a reciprocal for division Reverse the divisor before multiplying. Both fractions must be defined, and the divisor must be nonzero \(\displaystyle \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c},\quad b,c,d\ne0\)
5. Use a common denominator For addition or subtraction, form one denominator before combining numerators. Both original denominators must be nonzero \(\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd},\quad b,d\ne0\)

Worked example

Simplify the algebraic fraction and state all excluded values

1. \(\displaystyle \frac{3\left(a+2\right)}{2\left(a-4\right)}\times\frac{5a}{6\left(a-4\right)}\)Change division to multiplication by reversing the divisor
2. \(\displaystyle \frac{3\times5}{2\times6}=\frac{5}{4}\)Reduce the numerical coefficient
3. \(\displaystyle \frac{5a\left(a+2\right)}{4\left(a-4\right)^{2}}\)Combine the two matching factors in the denominator
4. \(\displaystyle \frac{5a\left(a+2\right)}{4\left(a-4\right)^{2}},\quad a\ne0,\ a\ne4\)Keep the restrictions from the original denominators, so a cannot be 0 or 4.

Common mistakes

Difficulty levels

Questions about this topic

In this chapter