Simplifying Algebraic Fractions
Cancelling algebraic fractions
Let us look at cancelling algebraic fractions first. We shall look at algebraic fractions that involve quadratic expressions.
Simplify the following fraction;
= 1/x + 2
We have to factorise the denominator of the fraction to simplify it first. This will make it easier to evaluate. We do so to get the following outcome.
We can see a common factor in both the numerator and the denominator so we can cancel these out;
= 1/x + 2
We have managed to workout the fraction, i.e: Simplify it.
Simplify the following fraction.
We need to find the common denominator first. We can do this simply by multiplying both denominators in the fraction.
Lastly we simplify by cleaning up the fraction of the multiplication signs. We get the answer as being.
Simplify the following fraction.
First we need to find the common denominator as we’ve done above, but before we do that we need to factorise the denominators as shown below;
You can see above that both fraction denominators have a factor of (x + 2) so we only include this once, because it will be cancelled out in the end anyway. You can try to include the extra (x + 2) and see what happens later. Remember what you multiply in the denominator must be multiplied with the numerator to make it fair otherwise the fractions won’t be equivalent. The lowest common denominator we find here is;
Now we must multiply the numerator as well. We multiply 2 by (x – 2) because to find the common denominator we multiplied by (x – 2)
For the fraction which has 5 to have the common denominator we must multiply 5 with (x + 3) as shown below;
Next we expand the brackets.
And lastly we simplify the fraction.
Multiplying and dividing algebraic fractions
In the following examples we’re going to look at multiplying and dividing algebraic fractions.
Simplify the following fraction.
First we check whether any numbers can cancel. Here 3 cannot cancel with 2 or 4 so we have to move on to the next step. Next we check the variables or letters. We see that the x in one fraction will cancel with one of the xs in the x3 of the other fraction. We can cancel these;
We can also cancel out x+7 from the numerator of the first fraction with the denominator if the second fraction as shown below;
That leaves;
Since we cannot cancel any more we simplify just multiply what is left.
The final answer then becomes;
The following example involves division.
Look at each numerator and denominator and see whether you can find their common factors. If you look carefully you will see that;
- 4x+12 has a common factor of 4
- 6x 2-12x -90 has common factor of 6
- 5x – 10 has a common factor of 5
- x2 – 4 has no common factors
We rewrite the expression as shown below;
Now we can factorise the quadratics in each fraction if any.
- x2-2x-15 factorises to (x+3)(x-5)
- x2-4 factorises to (x+2)(x-2)
Now we have the fraction as;
Next we turn the second fraction upside down and change the division sign to multiplication as shown below.
Now we can cancel out the common factors. We can divide 4 and 6 by 2.
We can also cancel out the factor of (x+3)
Also (x-2) can be cancelled out.
We can’t cancel out any more which leaves;
Now we can multiply what is left together. This leaves;
…as the final simplified answer.
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why are there no expressions that show subtraction. It moves from addition to multiplication to division
Thanks for the article… It helped me a lot
Yeah Bro this is very useful to those who are new to simplifying algebraic fractions. Thanks a lot mate, id rate this an 8! GOOD JOB KEEP ON COMING WITH THE GOOD WORK BUDS! SwAgmEiSter free styling out! Peace out Bruh’s!