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Reducing rational expressions to lowest terms

Learn what it means to reduce a rational expression to lowest terms, and how it's done!

What you should be familiar with before taking this lesson

A rational expression is a ratio of two polynomials. The domain of a rational expression is all real numbers except those that make the denominator equal to zero.
For example, the domain of the rational expression x+2x+1 is all real numbers except -1, or x1.
If this is new to you, we recommend that you check out our intro to rational expressions.
You should also know how to factor polynomials for this lesson.

What you will learn in this lesson

In this article, we will learn how to reduce rational expressions to lowest terms by looking at several examples.

Introduction

A rational expression is reduced to lowest terms if the numerator and denominator have no factors in common.
We can reduce rational expressions to lowest terms in much the same way as we reduce numerical fractions to lowest terms.
For example, 68 reduced to lowest terms is 34. Notice how we canceled a common factor of 2 from the numerator and the denominator:
68=2324=2324=34

Example 1: Reducing x2+3xx2+5x to lowest terms

Step 1: Factor the numerator and denominator
The only way to see if the numerator and denominator share common factors is to factor them!
x2+3xx2+5x=x(x+3)x(x+5)
Step 2: List restricted values
At this point, it is helpful to notice any restrictions on x. These will carry over to the simplified expression.
Since division by 0 is undefined, here we see that x0 and x5.
x(x+3)x(x+5)
Step 3: Cancel common factors
Now notice that the numerator and denominator share a common factor of x. This can be canceled out.
x(x+3)x(x+5)=x(x+3)x(x+5)=x+3x+5
Step 4: Final answer
Recall that the original expression is defined for x0,5. The reduced expression must have the same restrictions.
Because of this, we must note that x0. We do not need to note that x5, since this is understood from the expression.
In conclusion, the reduced form is written as follows:
x+3x+5 for x0

A note about equivalent expressions

Original expressionReduced expression
x2+3xx2+5xx+3x+5 for x0
The two expressions above are equivalent. This means their outputs are the same for all possible x-values. The table below illustrates this for x=2.
Original expressionReduced expression
Evaluation at x=2(2)2+3(2)(2)2+5(2)=1014=2527=2527=572+32+5=57=57=57=57
NoteThe result is reduced to lowest terms by canceling a common factor of 2.The result is already reduced to lowest terms because the factor of x (in this case x=2), was already canceled when we reduced the expression to lowest terms.
For this reason, the two expressions have the same value for the same input. However, values that make the original expression undefined often break this rule. Notice how this is the case with x=0.
Original expressionReduced expression (without restriction)
Evaluation at x=0(0)2+3(0)(0)2+5(0)=00=undefined0+30+5=35undefined
Because the two expressions must be equivalent for all possible inputs, we must require x0 for the reduced expression.

Misconception alert

Note that we cannot cancel the x's in the expression below. This is because these are terms rather than factors of the polynomials!
x+3x+535
This becomes clear when looking at a numerical example. For example, suppose x=2.
2+32+535
As a rule, we can only cancel if the numerator and denominator are in factored form!

Summary of the process for reducing to lowest terms

  • Step 1: Factor the numerator and the denominator.
  • Step 2: List restricted values.
  • Step 3: Cancel common factors.
  • Step 4: Reduce to lowest terms and note any restricted values not implied by the expression.

Check your understanding

Problem 1
Reduce 6x+202x+10 to lowest terms.
صرف 1 جواب چنو

Problem 2
Reduce x33x24x25x to lowest terms.
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

Example 2: Reducing x29x2+5x+6 to lowest terms

Step 1: Factor the numerator and denominator
x29x2+5x+6=(x3)(x+3)(x+2)(x+3)
Step 2: List restricted values
Since division by 0 is undefined, here we see that x2 and x3.
(x3)(x+3)(x+2)(x+3)
Step 3: Cancel common factors
Notice that the numerator and denominator share a common factor of x+3. This can be canceled out.
(x3)(x+3)(x+2)(x+3)=(x3)(x+3)(x+2)(x+3)=x3x+2
Step 4: Final answer
We write the reduced form as follows:
x3x+2 for x3
The original expression requires x2,3. We do not need to note that x2, since this is understood from the expression.

Check for understanding

Problem 3
Reduce x23x+2x21 to lowest terms.
صرف 1 جواب چنو

Problem 4
Reduce x22x15x2+x6 to lowest terms.
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