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Lesson Explainer: Simplifying Algebraic Fractions Mathematics • 10th Grade

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 3๐‘ฅ+4๐‘ฅโˆ’๐‘ฅ๐‘ฅ๏Šฉ๏Šจ and ๐‘ฅ+3๐‘ฅ+2๐‘ฅ+5๐‘ฅ+4๏Šจ๏Šจ.

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 3๐‘ฅ+4๐‘ฅโˆ’๐‘ฅ๐‘ฅ.๏Šฉ๏Šจ

The first step is to rewrite it as three separate algebraic fractions, each consisting of a monomial divided by a monomial: 3๐‘ฅ๐‘ฅ+4๐‘ฅ๐‘ฅโˆ’๐‘ฅ๐‘ฅ.๏Šฉ๏Šจ

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 ๐‘ฅ=๐‘ฅร—๐‘ฅร—๐‘ฅ๏Šฉ, we see that the ๐‘ฅ๏Šฉ on the top cancels with the ๐‘ฅ on the bottom to give ๐‘ฅ๏Šจ. In the second term, the ๐‘ฅ=๐‘ฅร—๐‘ฅ๏Šจ on the top cancels with the ๐‘ฅ on the bottom to give ๐‘ฅ; and in the third term, the ๐‘ฅ on the top cancels with the ๐‘ฅ on the bottom to give 1: 3ร—๐‘ฅร—๐‘ฅร—๐‘ฅ๐‘ฅ+4ร—๐‘ฅร—๐‘ฅ๐‘ฅโˆ’๐‘ฅ๐‘ฅ=3๐‘ฅ+4๐‘ฅโˆ’1.๏Šจ

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 ๐‘ฅ๐‘ฅ=๐‘ฅ๏Œบ๏Œป๏Œบ๏Šฑ๏Œป. Using this rule and remembering that ๐‘ฅ=๐‘ฅ๏Šง and ๐‘ฅ=1๏Šฆ, we get 3๐‘ฅ๐‘ฅ+4๐‘ฅ๐‘ฅโˆ’๐‘ฅ๐‘ฅ=3๐‘ฅ๐‘ฅ+4๐‘ฅ๐‘ฅโˆ’๐‘ฅ๐‘ฅ=3๐‘ฅ+4๐‘ฅโˆ’๐‘ฅ=3๐‘ฅ+4๐‘ฅโˆ’๐‘ฅ=3๐‘ฅ+4๐‘ฅโˆ’1.๏Šฉ๏Šจ๏Šฉ๏Šง๏Šจ๏Šง๏Šง๏Šง(๏Šฉ๏Šฑ๏Šง)(๏Šจ๏Šฑ๏Šง)(๏Šง๏Šฑ๏Šง)๏Šจ๏Šง๏Šฆ๏Šจ

As expected, this gives the same result as before.

In reality, rather than writing out powers of ๐‘ฅ 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: 3๐‘ฅ๐‘ฅ+4๐‘ฅ๐‘ฅโˆ’๐‘ฅ๐‘ฅ=3๐‘ฅ๐‘ฅ+4๐‘ฅ๐‘ฅโˆ’๐‘ฅ๐‘ฅ=3๐‘ฅ+4๐‘ฅโˆ’1.๏Šฉ๏Šจ๏Šจ๏Žข๏Žก๏Žก๏Ž 

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 9๐‘ฅโˆ’15๐‘ฅ+๐‘ฅ3๐‘ฅ๏Šฎ๏Šจ.

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 9๐‘ฅโˆ’15๐‘ฅ+๐‘ฅ3๐‘ฅ=9๐‘ฅ3๐‘ฅโˆ’15๐‘ฅ3๐‘ฅ+๐‘ฅ3๐‘ฅ.๏Šฎ๏Šจ๏Šฎ๏Šจ

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: 9๐‘ฅ3๐‘ฅโˆ’15๐‘ฅ3๐‘ฅ+๐‘ฅ3๐‘ฅ=3๐‘ฅ๐‘ฅโˆ’5๐‘ฅ๐‘ฅ+๐‘ฅ3๐‘ฅ.๏Šฉ๏Šฎ๏Šซ๏Šจ๏Šฎ๏Šจ

Finally, we cancel out the variables, which in this case means canceling the common factor ๐‘ฅ in all three fractions. This gives us the result 3๐‘ฅ๐‘ฅโˆ’5๐‘ฅ๐‘ฅ+๐‘ฅ3๐‘ฅ=3๐‘ฅโˆ’5๐‘ฅ+13.๏Žง๏Žก๏Žฆ๏Ž ๏Šญ

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 ๐‘ฅ(๐‘ฅ+5)(๐‘ฅโˆ’1)(๐‘ฅโˆ’1)(๐‘ฅ+6).

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 (๐‘ฅโˆ’1) appears in both the numerator and the denominator, we cancel out this common factor to get ๐‘ฅ(๐‘ฅ+5)(๐‘ฅโˆ’1)(๐‘ฅโˆ’1)(๐‘ฅ+6)=๐‘ฅ(๐‘ฅ+5)๐‘ฅ+6.

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 (๐‘ฅ+4)(๐‘ฅ+3)(๐‘ฅ+2)(๐‘ฅ+4).

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 (๐‘ฅ+4)(๐‘ฅ+3)(๐‘ฅ+2)(๐‘ฅ+4).

Notice that (๐‘ฅ+4) appears in both the numerator and the denominator, so we can cancel out this common factor to get (๐‘ฅ+4)(๐‘ฅ+3)(๐‘ฅ+2)(๐‘ฅ+4)=๐‘ฅ+3๐‘ฅ+2.

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 (๐‘ฅโˆ’2)(๐‘ฅโˆ’2)(๐‘ฅ+2)๏Šจ.

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 (๐‘ฅโˆ’2)(๐‘ฅโˆ’2)(๐‘ฅ+2).๏Šจ

We can rewrite the numerator (๐‘ฅโˆ’2)๏Šจ as (๐‘ฅโˆ’2)(๐‘ฅโˆ’2), so our algebraic fraction becomes (๐‘ฅโˆ’2)(๐‘ฅโˆ’2)(๐‘ฅโˆ’2)(๐‘ฅ+2).

Now, notice that the factor (๐‘ฅโˆ’2) 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 (๐‘ฅโˆ’2)(๐‘ฅโˆ’2)(๐‘ฅโˆ’2)(๐‘ฅ+2)=๐‘ฅโˆ’2๐‘ฅ+2, which is our final answer.

Alternatively, we could have canceled out the common factor (๐‘ฅโˆ’2) directly, as follows: (๐‘ฅโˆ’2)(๐‘ฅโˆ’2)(๐‘ฅ+2)=๐‘ฅโˆ’2๐‘ฅ+2.๏Žก๏Ž 

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 ๐‘ฅ+3๐‘ฅ+2๐‘ฅ+5๐‘ฅ+4.๏Šจ๏Šจ

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 ๐‘ฅ+3๐‘ฅ+2=(๐‘ฅ+1)(๐‘ฅ+2)๏Šจ. Similarly, we can factor the denominator to obtain ๐‘ฅ+5๐‘ฅ+4=(๐‘ฅ+1)(๐‘ฅ+4)๏Šจ. Therefore, we have the factored version (๐‘ฅ+1)(๐‘ฅ+2)(๐‘ฅ+1)(๐‘ฅ+4).

As (๐‘ฅ+1) appears in both the numerator and the denominator, we can cancel this common factor out to get (๐‘ฅ+1)(๐‘ฅ+2)(๐‘ฅ+1)(๐‘ฅ+4)=๐‘ฅ+2๐‘ฅ+4.

Let us now try an example to test this skill.

Example 4: Simplifying an Algebraic Expression by Factoring the Numerator and Denominator

Fully simplify ๐‘ฅ+5๐‘ฅโˆ’24๐‘ฅ+15๐‘ฅ+56๏Šจ๏Šจ.

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, ๐‘ฅ+5๐‘ฅโˆ’24๏Šจ, we need to identify factor pairs that multiply to give โˆ’24 and then select the one that adds to give 5. It is simple to check that the required numbers are โˆ’3 and 8, so we can factor the numerator to get (๐‘ฅโˆ’3)(๐‘ฅ+8).

Similarly, to factor the denominator, ๐‘ฅ+15๐‘ฅ+56๏Šจ, 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 (๐‘ฅ+7)(๐‘ฅ+8).

Therefore, we have the factored version (๐‘ฅโˆ’3)(๐‘ฅ+8)(๐‘ฅ+7)(๐‘ฅ+8).

Notice that (๐‘ฅ+8) appears in both the numerator and the denominator, so we can cancel this common factor out to get (๐‘ฅโˆ’3)(๐‘ฅ+8)(๐‘ฅ+7)(๐‘ฅ+8)=๐‘ฅโˆ’3๐‘ฅ+7.

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

  1. 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.
  2. 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.
  3. 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 4๐‘ฅโˆ’16๐‘ฅโˆ’20๐‘ฅ2๐‘ฅโˆ’16๐‘ฅ+30=๐‘Ž๐‘ฅ(๐‘ฅ+๐‘)๐‘ฅ+๐‘๏Šฉ๏Šจ๏Šจ, where ๐‘Ž, ๐‘, and ๐‘ are integers. Work out the values of ๐‘Ž, ๐‘, and ๐‘.

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 ๐‘Ž, ๐‘, and ๐‘.

Starting with the numerator, 4๐‘ฅโˆ’16๐‘ฅโˆ’20๐‘ฅ๏Šฉ๏Šจ, note that each term has 4๐‘ฅ as a factor, so we can take out this common factor to get 4๐‘ฅ๏€น๐‘ฅโˆ’4๐‘ฅโˆ’5๏….๏Šจ

What remains inside the brackets is the quadratic expression ๐‘ฅโˆ’4๐‘ฅโˆ’5๏Šจ, which we need to factor. To do this, we must identify factor pairs that multiply to give โˆ’5 and then select the one that adds to give โˆ’4. It is simple to check that the required numbers are โˆ’5 and 1, so we can factor this quadratic expression to get (๐‘ฅโˆ’5)(๐‘ฅ+1). Hence, our fully factored numerator is 4๐‘ฅ(๐‘ฅโˆ’5)(๐‘ฅ+1).

Turning our attention to the denominator, 2๐‘ฅโˆ’16๐‘ฅ+30๏Šจ, each term has 2 as a factor. Taking out this common factor, we get 2๏€น๐‘ฅโˆ’8๐‘ฅ+15๏….๏Šจ

Inside the brackets, we have the quadratic expression ๐‘ฅโˆ’8๐‘ฅ+15๏Šจ, 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 โˆ’8. It is simple to check that the required numbers are โˆ’5 and โˆ’3, so we can factor this quadratic expression to get (๐‘ฅโˆ’5)(๐‘ฅโˆ’3). Therefore, our fully factored denominator is 2(๐‘ฅโˆ’5)(๐‘ฅโˆ’3).

We can then reassemble our original algebraic fraction in its factored form: 4๐‘ฅ(๐‘ฅโˆ’5)(๐‘ฅ+1)2(๐‘ฅโˆ’5)(๐‘ฅโˆ’3).

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 4๐‘ฅ(๐‘ฅโˆ’5)(๐‘ฅ+1)2(๐‘ฅโˆ’5)(๐‘ฅโˆ’3)=2๐‘ฅ(๐‘ฅโˆ’5)(๐‘ฅ+1)(๐‘ฅโˆ’5)(๐‘ฅโˆ’3).๏Šจ

Then, looking at the variable terms, notice that (๐‘ฅโˆ’5) appears in both the numerator and the denominator, so we can cancel this common factor out to get 2๐‘ฅ(๐‘ฅโˆ’5)(๐‘ฅ+1)(๐‘ฅโˆ’5)(๐‘ฅโˆ’3)=2๐‘ฅ(๐‘ฅ+1)๐‘ฅโˆ’3.

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 ๐‘Ž, ๐‘, and ๐‘. Thus, we have 2๐‘ฅ(๐‘ฅ+1)๐‘ฅโˆ’3=๐‘Ž๐‘ฅ(๐‘ฅ+๐‘)๐‘ฅ+๐‘, from which it follows that ๐‘Ž=2, ๐‘=1, and ๐‘=โˆ’3.

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.

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