Simplify an algebraic fraction by factorising and cancelling practice questions | Year 10 Mathematics

Cancel factors without changing the domain

Simplifying changes the written rule. It does not add inputs to the original domain

Record the original domain Every zero of the original denominator stays excluded, whether its factor cancels or remains
Factor whole expressions Write sums as products before cancelling. A common factor must multiply the entire numerator and denominator \(\displaystyle x^2+2x-8=(x-2)(x+4)\)
Cancel matching copies Cancel equal factors only when they are nonzero. Repeated factors cancel one matching copy at a time \(\displaystyle \begin{aligned}&\frac{(x-2)^2}{x-2}\\&=x-2,\quad x\ne2\end{aligned}\)
Keep the original restrictions The simpler rule represents the original fraction only on its original domain \(\displaystyle \begin{aligned}&\frac{x^2+2x-8}{x-2}\\&=x+4,\quad x\ne2\end{aligned}\)

Simplify and retain every original excluded value

\frac{x-2}{x^2+x-6}

\(\displaystyle \begin{aligned}&x^2+x-6\\&=(x-2)(x+3)\end{aligned}\)Factor the original denominator
\(\displaystyle x=2\quad\text{or}\quad x=-3\)These values make that denominator zero
\(\displaystyle x\ne2,\quad x\ne-3\)Exclude both values before cancellation
\(\displaystyle \frac{x-2}{(x-2)(x+3)}\)Replace the denominator with its factors
\(\displaystyle \frac{1}{x+3}\)Cancel one matching nonzero factor. Both restrictions remain

Cancelling terms

✕\(\displaystyle \frac{x+2}{x+5}=\frac25\)✓\(\displaystyle \frac{1+2}{1+5}=\frac12\)

The allowed counterexample x = 1 disproves the cancellation. Cancel whole common factors only

Losing a cancelled restriction

✕\(\displaystyle \begin{aligned}&\frac{x^2-4}{x-2}\\&=x+2\quad\text{for all }x\end{aligned}\)✓\(\displaystyle x+2,\quad x\ne2\)

The original denominator is zero at 2.

Cancelling too many copies

✕\(\displaystyle \frac{(x-3)^2}{x-3}=1\)✓\(\displaystyle x-3,\quad x\ne3\)

Only one of the two numerator copies cancels

introFactor a numerator, cancel a monomial or binomial and keep its restriction

standardFactor a denominator and retain all of its distinct zeros as exclusions

challengeRepeated factors and differences of squares in either orientation

examChoose the complete simplified expression and original domain across all patterns

Why is zero divided by zero not zero?+

Division asks for a unique multiplier. Every number times zero is zero, so the expression below does not specify a unique result and is undefined

\frac00

Can a cancelled input be restored?+

Not when simplifying the original expression. You could define a new extended function, but it would have a different domain

Does a repeated denominator root count twice?+

List the excluded value once. Multiplicity affects cancellation, not the set of allowed inputs

Can the numerator be zero?+

Yes, if the original denominator is nonzero. A zero numerator alone is allowed and gives value zero

What if the brackets have opposite signs?+

Rewrite one using a factor of −1 and keep that sign when cancelling

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs