E2.3 - Calculating with algebraic fractions

Factorising and simplifying rational expressions

Is $\frac{x^2-9}{x-3}$ really the same as $x+3$? Factoring the numerator gives $\frac{(x-3)(x+3)}{x-3}$, so cancellation looks promising. But at $x=3$, the original denominator is zero. The simplified expression can preserve the value only where the original was defined.

Finding that simple exclusion needs no new equation-solving machinery: $x-2$ is zero exactly when $x$ is $2$, because $2-2=0$. Similarly, $x+3$ is zero when $x=-3$. A product is zero when at least one factor is zero, so factor a denominator to see its forbidden inputs.

Cancellation divides numerator and denominator by a common factor, a whole multiplied piece. For nonzero $F$, $\frac{FG}{FH}=\frac GH$, provided $H\ne0$ too. It cannot remove one term from an addition: $\frac{x+3}{x}$ is not $3$.

Change only the denominator, adding a factor $x-1$. That factor stays below the line, and it adds another excluded value.

The opening expression therefore simplifies to $x+3$ for $x\ne3$. It does not acquire a new value at the excluded input just because the shorter formula has one.

A factor that cancels still contributes its original excluded value in the next fraction.

Products and quotients of algebraic fractions

For multiplication, multiply numerators together and denominators together. Factorising first often lets you cancel before expanding anything. Numerical denominators need no special rule: $\frac{2a}{3}\times\frac{9a}{4}=\frac{18a^2}{12}=\frac{3a^2}{2}$.

Division means multiplication by the reciprocal of the whole divisor. The divisor must be defined and nonzero. Consequently, its original denominator cannot be zero, and neither can its numerator. Write these restrictions before turning it upside down; otherwise an original denominator restriction may disappear from view.

Change only the operation to division. The second fraction is now a divisor, so $x=3$ must also be excluded: it would make that divisor zero.

For comparison, $\frac{2a}{3}\div\frac{9a}{4}=\frac{2a}{3}\times\frac4{9a}=\frac8{27}$, but only for $a\ne0$. The original divisor is zero at $a=0$, even though the simplified answer is a constant.

Before inverting the divisor, record the input that would make its numerator zero as well as the original denominator restrictions.

Sums and differences with common denominators

You can add thirds to halves only after expressing them in equal-sized pieces. The same is true when the numerators contain letters. Multiply a fraction's numerator and denominator by the same nonzero factor to create an equivalent fraction. Then add or subtract the whole numerators, keeping the denominator.

For numerical denominators, use a common multiple. For denominators such as $x-2$ and $x-3$, their product is a common denominator. If denominators share a factor, include it only as often as needed. Restrictions come from the original denominators.