Cancel the common factor
Simplifying rational expressions
A fraction in x
A fraction such as 69 becomes simpler when we cancel the common factor 3: we get the simpler fraction 23 in its place. An algebra fraction, called a rational expression, can be simplified in the same way. But the top and the bottom are sums, so we cannot cancel yet.
What do we have to do first?
What this lesson covers
The idea
A rational algebraic expression is simplified by factorising its numerator and denominator and cancelling their common factors, provided such a factor is not equal to zero.
A fraction in x
A fraction such as 6/9 becomes simpler when we cancel the common factor 3: we get the simpler fraction 2/3 in its place. An algebra fraction, called a rational expression, can be simplified in the same way. But the top and the bottom are sums, so we cannot cancel yet.
What do we have to do first?
- Factorise the top and the bottom
- Cancel the x² terms
- Multiply the top and the bottom by x
Cancel the factors
Each expression is already factorised. Tap a factor on the top, then the same factor on the bottom, to cancel it. Then slide x to compare the original with the simplified expression.
Factorise, then cancel
A rational expression is simplified by factorising its top and bottom and cancelling the common factors, provided such a factor is not equal to zero.
x² − 7x + 12 = (x − 3)(x − 4), and 5x² + 5x − 100 = 5(x − 4)(x + 5). The factor x − 4 is on top and at the bottom, so it cancels, as long as it is not zero. The result is (x − 3) over 5(x + 5).
Notes
Factorise the top and the bottom, then cancel the common factors, provided such a factor is not zero.
Check yourself
In (x + 2)(x + 3) over (x + 1)(x + 2), which factor can be cancelled?
(x + 2)(x + 3) over (x + 1)(x + 2) becomes (x + 3) over (x + 1) after cancelling x + 2. For which value of x are we not allowed to cancel x + 2, because it is zero?
Answer: -2
x + 2 = 0 when x = −2. At x = −2 the original expression has 0 at the bottom, so it is not defined.
Simplify x² + 5x + 6 over x + 2.
In the example above, the factor x − 4 was cancelled. For which value of x would cancelling it be wrong, because x − 4 = 0?
Answer: 4
x − 4 = 0 when x = 4. That is why the book says the expression is simplified provided 5x² + 5x − 100 ≠ 0.
- x + 3. x + 3 is only on the top, so it has no partner at the bottom.
- x + 2 — correct. Yes. It is on the top and on the bottom.
- x + 1. x + 1 is only on the bottom, so it has no partner on the top.
- x² + 5x + 3. The 2 in x + 2 is a term, not a factor of the top. Factorise first: x² + 5x + 6 = (x + 2)(x + 3).
- (x + 2)(x + 3). That is the top itself. Nothing has been cancelled.
- x + 3 — correct. Yes: x² + 5x + 6 = (x + 2)(x + 3), and x + 2 cancels.