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Algebra (all content)

Course: Algebra (all content) > Unit 13

Lesson 2: Simplifying rational expressions
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Algebra (all content)
Rational expressions, equations, & functions
Simplifying rational expressions
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Simplifying rational expressions (advanced)

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Have you learned the basics of rational expression simplification? Great! Now gain more experience with some trickier examples.

What you should be familiar with before taking this lesson

A rational expression is a ratio of two polynomials. A rational expression is considered simplified if the numerator and denominator have no factors in common.
If this is new to you, we recommend that you check out our intro to simplifying rational expressions.

What you will learn in this lesson

In this lesson, you will practice simplifying more complicated rational expressions. Let's look at two examples, and then you can try some problems!

Example 1: Simplifying  10x32x2−18x

Step 1: Factor the numerator and denominator
Here it is important to notice that while the numerator is a monomial, we can factor this as well.
10x32x2−18x=2⋅5⋅x⋅x22⋅x⋅(x−9)
Step 2: List restricted values
From the factored form, we see that x≠0 and x≠9.
Step 3: Cancel common factors
2⋅5⋅x⋅x22⋅x⋅(x−9)=2⋅5⋅x⋅x22⋅x⋅(x−9)=5x2x−9
Step 4: Final answer
We write the simplified form as follows:
5x2x−9 for x≠0

Main takeaway

In this example, we see that sometimes we will have to factor monomials in order to simplify a rational expression.

Check your understanding

1) Simplify 6x212x4−9x3.
Choose 1 answer:

Example 2: Simplifying  (3−x)(x−1)(x−3)(x+1)

Step 1: Factor the numerator and denominator
While it does not appear that there are any common factors, x−3 and 3−x are related. In fact, we can factor −1 out of the numerator to reveal a common factor of x−3.
=(3−x)(x−1)(x−3)(x+1)=−1(−3+x)(x−1)(x−3)(x+1)=−1(x−3)(x−1)(x−3)(x+1)Commutativity
Step 2: List restricted values
From the factored form, we see that x≠3 and x≠−1.
Step 3: Cancel common factors
=−1(x−3)(x−1)(x−3)(x+1)=−1(x−3)(x−1)(x−3)(x+1)=−1(x−1)x+1=1−xx+1
The last step of multiplying the −1 into the numerator wasn't necessary, but it is common to do so.
Step 4: Final answer
We write the simplified form as follows:
1−xx+1 for x≠3

Main takeaway

The factors x−3 and 3−x are opposites since −1⋅(x−3)=3−x.
In this example, we saw that these factors canceled, but that a factor of −1 was added. In other words, the factors x−3 and 3−x canceled to -1.
In general opposite factors a−b and b−a will cancel to −1 provided that a≠b.

Check your understanding

2) Simplify (x−2)(x−5)(2−x)(x+5).
Choose 1 answer:

3) Simplify 15−10x8x3−12x2.
for x≠
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Let's try some more problems

4) Simplify 3x15x2−6x.
Choose 1 answer:

5) Simplify 3x3−15x2+12x3x−3.
for x≠
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

6) Simplify 6x2−12x6x−3x2.
Choose 1 answer:

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  • piceratops seedling style avatar for user cobri17
    Posted 7 years ago. Direct link to cobri17's post “I'm completely lost on Ra...”
    I'm completely lost on Rational expressions. I know how to factor, however I don't understand the whole restriction thing.
    (8 votes)
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    • aqualine tree style avatar for user Judith Gibson
      Posted 7 years ago. Direct link to Judith Gibson's post “If we divide out a common...”
      If we divide out a common factor from both numerator and denominator so that it disappears entirely from the fraction, then we need to restrict x from being whatever value made that factor equal zero.
      For example, ( x + 3 ) ( x - 5 ) / (x + 3) = ( x + 5 ) but x can't equal -3.
      For example, ( y - 7 ) ( y + 2 ) / ( y - 7 ) ^2 = ( y + 2 ) / ( y - 7 ), but there's no need to list any restrictions for y because even though one factor of ( y - 7 ) has been divided out, a factor of ( y - 7 ) still remains in the denominator.
      (7 votes)
  • blobby green style avatar for user khare9770
    Posted 7 years ago. Direct link to khare9770's post “3x+1/x : express as ratio...”
    3x+1/x : express as rational expression
    (6 votes)
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  • blobby green style avatar for user Fred Haynes
    Posted 3 years ago. Direct link to Fred Haynes's post “I just finished an the Pr...”
    I just finished an the Practice Problems in the next exercise. To get to the part I don't understand I will simply ask this question which was the last section of the problem.

    The final answer was -(z+11)/(z-4) then it was multiplied out to z+11/4-z. Why was the answer -(z+11)/(z-4) not correct.

    Thanks
    (3 votes)
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  • spunky sam blue style avatar for user mohammad ismail memon
    Posted 5 years ago. Direct link to mohammad ismail memon's post “it is pretty hard and spe...”
    it is pretty hard and specially the one where you have to choose 3 answers
    (4 votes)
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  • mr pink red style avatar for user AndyRupnarine
    Posted 8 years ago. Direct link to AndyRupnarine's post “For question 3, I don't u...”
    For question 3, I don't understand why the answer is -5/4x^2 for x =/= 3/2. Why not x=/= 0 as well? I got the result (-5/4x^2) correct but when I tried to enter for x=/= "0, 3/2" I keep getting, "we couldn't understand your answer."
    (2 votes)
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  • aqualine ultimate style avatar for user Tryphosa Thatheu
    Posted 5 years ago. Direct link to Tryphosa Thatheu's post “(x−3)(x+6) x 2 −16 ​ ÷ x ...”
    (x−3)(x+6) x 2 −16 ​ ÷ x 2 +cx−18 (x+a)(a−x) ​ =−1

    I want to know what c is..
    (1 vote)
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  • blobby green style avatar for user MARCKENLEY
    Posted 2 years ago. Direct link to MARCKENLEY's post “Determine whether 3^{2m}\...”
    Determine whether 3^{2m}\cdot 4^{2m}3
    2m
    ⋅4
    2m
    3, start superscript, 2, m, end superscript, dot, 4, start superscript, 2, m, end superscript is equivalent to each of the following expressions.
    (2 votes)
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  • aqualine ultimate style avatar for user 007euclidd
    Posted 5 years ago. Direct link to 007euclidd's post “shouldn't (3-2x) and (2x-...”
    shouldn't (3-2x) and (2x-3) simplify to (+1) when divided, not (-1) because they are both negative already?
    (1 vote)
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  • blobby green style avatar for user mntambonondumiso24
    Posted 7 years ago. Direct link to mntambonondumiso24's post “How do you simplify (1/x...”
    How do you simplify (1/x)/1/2rootx
    (1 vote)
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    • cacteye blue style avatar for user Jerry Nilsson
      Posted 7 years ago. Direct link to Jerry Nilsson's post “_If_ this expression is c...”
      If this expression is correctly written, then the order of operations (PEMDAS) tells us to first simplify 1/𝑥, which cannot be simplified any further, then √𝑥 (because this is technically an exponent of 𝑥), which can neither be simplified any further, which means that we keep these expressions as they are.
      Next is multiplication/division, which we perform from left to right:
      (1/𝑥)/1 = 1/𝑥.
      (1/𝑥)/2 = 1/(2𝑥).
      (1/(2𝑥)) ∙ √𝑥 = √𝑥/(2𝑥) = 1/(2√𝑥)
      And we're done.

      – – –

      However, I have a hunch that you meant to simplify (1/𝑥)/(1/(2√𝑥)), where we can't simplify any of the parentheses or exponents, but we do have a fraction of the type (1/𝑎)/(1/𝑏), which simplifies as 𝑏/𝑎.
      So, (1/𝑥)/(1/(2√𝑥)) = 2√𝑥/𝑥 = 2/√𝑥.

      – – –

      Note that, in both examples, the domain of the simplified expression is the same as the domain of the original expression, which is not always the case. If the domain is somehow changed during the simplification process, we should include the domain of the original expression in our answer.
      (3 votes)
  • leaf blue style avatar for user Igor Skoldin
    Posted 6 years ago. Direct link to Igor Skoldin's post “What would happen if in t...”
    What would happen if in the Example 2 we multiply a term in the denominator (instead of a term in the nominator) to get a common factor?

    (3-x)(x-1)/(x-3)(x+1)
    = (3-x)(x-1)/(-1)(3-x)(x+1) # multiply (x-3) by -1 and get the common factor of (3-x)
    = (x-1)/(1+x) # cancel out (3-x) and multiply (x+1) by -1 in the denominator

    So we get (x-1)/(1+x) for x ≠ 3 (and x ≠ -1, which is implied by the expression). But we should have gotten (1-x)/(1+x), which is not exactly the same. Actually, it is equivalent to (x-1)(-1)/(1+x).

    Did I make a mistake or why doesn't it work?
    (1 vote)
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