In this explainer, we will learn how to factor algebraic expressions and simplify algebraic fractions.
Let us start with the definition of an algebraic fraction.
Definition: Algebraic Fraction
An algebraic fraction is a fraction that has algebraic expressions in the numerator and/or the denominator.
Two examples are and .
Note that the first example has a polynomial in the numerator and a single term (sometimes called a monomial) in the denominator, whereas the second has a polynomial in both the numerator and the denominator.
Since the easiest type of algebraic fractions to simplify involve polynomials divided by monomials, we will start by describing what to do in this case. Therefore, consider the algebraic fraction
The first step is to rewrite it as three separate algebraic fractions, each consisting of a monomial divided by a monomial:
Next, we cancel out any common factors on the top and bottom of the fractions. Notice that in this case, there are no constants to cancel, so we focus on the variables.
In the first term, recalling that , we see that the on the top cancels with the on the bottom to give . In the second term, the on the top cancels with the on the bottom to give ; and in the third term, the on the top cancels with the on the bottom to give 1:
Another way to think of this process is as three separate applications of the quotient rule of exponents. Recall that this rule states that the quotient of two powers of the same nonzero base satisfies . Using this rule and remembering that and , we get
As expected, this gives the same result as before.
In reality, rather than writing out powers of as repeated products or explicitly quoting the quotient rule of exponents, it is perfectly acceptable to simplify algebraic fractions by canceling from the top and the bottom. For example, once we had separated the original expression into individual algebraic fractions, we could have shown our working as follows:
Note that if there had also been constants to cancel, the process would have involved a simple additional step. To make this point completely clear, let us try the next example.
Example 1: Simplifying an Algebraic Fraction with a Single Term or a Monomial in the Denominator
Fully simplify the fraction .
Answer
Recall that to simplify an algebraic fraction with a polynomial in the numerator and a single term (called a monomial) in the denominator, we split it into separate algebraic fractions, each consisting of a monomial divided by a monomial. Then, we cancel out any common factors on the top and bottom of the fractions.
In this case, we convert the given expression into three separate algebraic fractions, so
Next, we cancel out any common factors on the top and bottom of the fractions. Starting with the constants, we cancel out the common factor of 3 in the first two fractions:
Finally, we cancel out the variables, which in this case means canceling the common factor in all three fractions. This gives us the result
Next, we turn our attention to simplifying algebraic fractions with polynomials in the numerator and denominator. The polynomials may appear in factored or unfactored form. For instance, consider the algebraic fraction
This expression has factored polynomials in both the numerator and the denominator. To simplify it, we cancel out any common factors on the top and bottom. Therefore, as appears in both the numerator and the denominator, we cancel out this common factor to get
As there are no further common factors to cancel out, this is our fully simplified answer.
Let us look at an example of this type.
Example 2: Simplifying a Factored Algebraic Expression
Fully simplify .
Answer
Recall that to simplify an algebraic fraction with factored polynomials in both the numerator and the denominator, we cancel out any common factors on the top and bottom.
In this case, we are asked to fully simplify the algebraic fraction
Notice that appears in both the numerator and the denominator, so we can cancel out this common factor to get
As no further cancelation is possible, this is our fully simplified answer.
Our next example builds upon this idea, as it requires a small additional step of working.
Example 3: Simplifying a Factored Algebraic Expression
Fully simplify .
Answer
Recall that to simplify an algebraic fraction with factored polynomials in both the numerator and the denominator, we cancel out any common factors on the top and bottom.
Here, we are asked to fully simplify the algebraic fraction
We can rewrite the numerator as , so our algebraic fraction becomes
Now, notice that the factor appears twice in the numerator and once in the denominator. Therefore, we can cancel out one copy from the numerator with the one from the denominator to get which is our final answer.
Alternatively, we could have canceled out the common factor directly, as follows:
As expected, this gives exactly the same answer.
The most complex algebraic fractions we will be asked to simplify are ones featuring unfactored polynomials. For instance, returning to the second example from our original definition, suppose we are asked to fully simplify the algebraic fraction
Our strategy will be to try to factor the polynomials in the numerator and the denominator. Once this is done, we can inspect the factored version of the algebraic fraction and, as previously, cancel out any common factors on the top and bottom.
In this case, notice that we can factor the numerator to obtain . Similarly, we can factor the denominator to obtain . Therefore, we have the factored version
As appears in both the numerator and the denominator, we can cancel this common factor out to get
Let us now try an example to test this skill.
Example 4: Simplifying an Algebraic Expression by Factoring the Numerator and Denominator
Fully simplify .
Answer
Recall that to simplify an algebraic fraction with any unfactored polynomials in the numerator and/or denominator, we first factor the polynomials. Then, we cancel out any common factors on the top and bottom.
In this question, the given algebraic fraction has unfactored quadratic expressions in both the numerator and the denominator.
To factor the numerator, , we need to identify factor pairs that multiply to give and then select the one that adds to give 5. It is simple to check that the required numbers are and 8, so we can factor the numerator to get .
Similarly, to factor the denominator, , we need to identify factor pairs that multiply to give 56 and then select the one that adds to give 15. In this case, the required numbers are 7 and 8, so we can factor the denominator to get .
Therefore, we have the factored version
Notice that appears in both the numerator and the denominator, so we can cancel this common factor out to get
As no further cancelation is possible, this is our fully simplified answer.
Let us now summarize what we have learned about simplifying different types of algebraic fractions.
How To: Simplifying Algebraic Fractions
- To simplify an algebraic fraction with a polynomial in the numerator and a single term (called a monomial) in the denominator, we split it into separate algebraic fractions, each consisting of a monomial divided by a monomial. Then, we cancel out any common factors on the top and bottom of the fractions.
- To simplify an algebraic fraction with factored polynomials in both the numerator and the denominator, we cancel out any common factors on the top and bottom.
- To simplify an algebraic fraction with any unfactored polynomials in the numerator and/or denominator, we first factor the polynomials. Then, we cancel out any common factors on the top and bottom.
By using the techniques discussed here, we can solve problems of greater complexity, as demonstrated in our final example.
Example 5: Finding Unknown Constants by Simplifying an Algebraic Expression
We are given that , where , , and are integers. Work out the values of , , and .
Answer
Recall that to simplify an algebraic fraction with any unfactored polynomials in the numerator and/or denominator, we first factor the polynomials. Then, we cancel out any common factors on the top and bottom.
Here, our strategy will be to take the algebraic fraction on the left-hand side of the given equation, factor the numerator and denominator, cancel out any common factors, and then compare the result with the algebraic fraction on the right-hand side of the given equation. We should then be able to read off the values of , , and .
Starting with the numerator, , note that each term has as a factor, so we can take out this common factor to get
What remains inside the brackets is the quadratic expression , which we need to factor. To do this, we must identify factor pairs that multiply to give and then select the one that adds to give . It is simple to check that the required numbers are and 1, so we can factor this quadratic expression to get . Hence, our fully factored numerator is
Turning our attention to the denominator, , each term has 2 as a factor. Taking out this common factor, we get
Inside the brackets, we have the quadratic expression , which we need to factor. To do this, we must identify factor pairs that multiply to give 15 and then select the one that adds to give . It is simple to check that the required numbers are and , so we can factor this quadratic expression to get . Therefore, our fully factored denominator is
We can then reassemble our original algebraic fraction in its factored form:
Next, we need to cancel out any common factors on the top and bottom of the fraction. Starting with the constants, we cancel out the common factor of 2, which gives
Then, looking at the variable terms, notice that appears in both the numerator and the denominator, so we can cancel this common factor out to get
Our final step is to set our fully factored algebraic fraction equal to the one on the right-hand side of the original equation and then read off the values of , , and . Thus, we have from which it follows that , , and .
Let us finish by recapping some key concepts from this explainer.
Key Points
- An algebraic fraction is a fraction that has algebraic expressions in the numerator and/or the denominator.
- To simplify an algebraic fraction with a polynomial in the numerator and a single term (called a monomial) in the denominator, we split it into separate algebraic fractions, each consisting of a monomial divided by a monomial. Then, we cancel out any common factors on the top and bottom of the fractions.
- To simplify an algebraic fraction with factored polynomials in both the numerator and the denominator, we cancel out any common factors on the top and bottom.
- To simplify an algebraic fraction with any unfactored polynomials in the numerator and/or denominator, we first factor the polynomials. Then, we cancel out any common factors on the top and bottom.
