Cancelling common factors to simplify rational expressions
Simplifying fractions using cancellation
In this lesson we’ll look at how to reduce rational expressions and how cancelling can be a useful technique to help us reduce them.
To reduce a fraction or a rational expression there must be a common factor in all of the parts that are being added or subtracted.
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A common factor is a number or term that all parts share. For example ???3x???, ???12x^2???, and ???9x^3??? all have a common factor of ???3x??? because, ???1\cdot 3x = 3x???, ???4x \cdot 3x = 12x^2??? and ???3x^2\cdot 3x= 9x^3???.
How to identify common factors in the numerator and denominator and cancel them in order to simplify the fraction
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Reducing the rational expression to its lowest terms
Example
Reduce the fraction to its lowest terms.
???\frac{9x^3-12x^2+3x}{15x^2}???
Let’s look for a common factor. We can factor a ???3x??? out of each term and then cancel it.
???\frac{3x\left(3x^2-4x+1\right)}{3x(5x)}???
???\frac{3x^2-4x+1}{5x}???
It would also work to take out each factor individually, first the ???3???, then the ???x???.
Let’s look at another example.
To reduce a fraction or a rational expression there must be a common factor in all of the parts that are being added or subtracted.
Example
Simplify each expression in the difference.
???\frac{12ab+8a^2b^2}{10ab}-\frac{36a^3b^3-6a^2b^2}{6ab}???
We want to factor everything that we can out of each numerator and denominator. That means we need to take every constant, every factor of ???a??? and every factor of ???b??? that's common to each term.
???\frac{12ab+8a^2b^2}{10ab}-\frac{36a^3b^3-6a^2b^2}{6ab}???
???\frac{2ab(6+4ab)}{2ab(5)}-\frac{6a^2b^2(6ab-1)}{6ab}???
???\frac{2ab(6+4ab)}{2ab(5)}-\frac{6ab(ab)(6ab-1)}{6ab}???
Then we want to cancel what's common between the numerator and denominator.
???\frac{6+4ab}{5}-\frac{(ab)(6ab-1)}{1}???
???\frac{6+4ab}{5}-ab(6ab-1)???
