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## Algebra (all content)

### Unit 13: Lesson 2

Simplifying rational expressions- Reducing rational expressions to lowest terms
- Reducing rational expressions to lowest terms
- Simplifying rational expressions: common monomial factors
- Reduce rational expressions to lowest terms: Error analysis
- Simplifying rational expressions: common binomial factors
- Simplifying rational expressions: opposite common binomial factors
- Simplifying rational expressions (advanced)
- Reduce rational expressions to lowest terms
- Simplifying rational expressions: grouping
- Simplifying rational expressions: higher degree terms
- Simplifying rational expressions: two variables
- Simplify rational expressions (advanced)
- Simplifying rational expressions (old video)

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# Reducing rational expressions to lowest terms

CCSS.Math:

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 start fraction, x, plus, 2, divided by, x, plus, 1, end fraction is

*all real numbers except start text, negative, 1, end text*, or x, does not equal, minus, 1.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, start fraction, 6, divided by, 8, end fraction reduced to lowest terms is start fraction, 3, divided by, 4, end fraction. Notice how we canceled a common factor of 2 from the numerator and the denominator:

## Example 1: Reducing start fraction, x, squared, plus, 3, x, divided by, x, squared, plus, 5, x, end fraction 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!

**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 start color #0c7f99, x, does not equal, 0, end color #0c7f99 and start color #7854ab, x, does not equal, minus, 5, end color #7854ab.

**Step 3: Cancel common factors**

Now notice that the numerator and denominator share a common factor of x. This can be canceled out.

**Step 4: Final answer**

Recall that the original expression is defined for x, does not equal, 0, comma, minus, 5. The reduced expression must have the same restrictions.

Because of this, we must note that x, does not equal, 0. We do not need to note that x, does not equal, minus, 5, since this is understood from the expression.

In conclusion, the reduced form is written as follows:

start fraction, x, plus, 3, divided by, x, plus, 5, end fraction for x, does not equal, 0

### A note about equivalent expressions

Original expression | Reduced expression |
---|---|

start fraction, x, squared, plus, 3, x, divided by, x, squared, plus, 5, x, end fraction | start fraction, x, plus, 3, divided by, x, plus, 5, end fraction for x, does not equal, 0 |

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, equals, 2.Original expression | Reduced expression | ||
---|---|---|---|

Evaluation at start color #7854ab, x, equals, 2, end color #7854ab | $\begin{aligned}\dfrac{(\purpleD{2})^2+3(\purpleD{2})}{(\purpleD{2})^2+5(\purpleD{2})}&=\dfrac{10}{14}\\\\&=\dfrac{\purpleD{{2}}\cdot 5}{\purpleD{{2}}\cdot 7}\\\\&=\dfrac{\purpleD{\cancel{2}}\cdot 5}{\purpleD{\cancel{2}}\cdot 7}\\\\&=\dfrac{5}{7}\end{aligned}$ | $\begin{aligned}\dfrac{\purpleD{2}+3}{\purpleD{2}+5}&=\dfrac{5}{7}\\\\&\phantom{=\dfrac57}\\\\&\phantom{=\dfrac57}\\\\&\phantom{=\dfrac57}\end{aligned}$ | |

Note | The result is reduced to lowest terms by canceling a common factor of start color #7854ab, 2, end color #7854ab. | The result is already reduced to lowest terms because the factor of x (in this case start color #7854ab, x, equals, 2, end color #7854ab), 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 start color #7854ab, x, equals, 0, end color #7854ab.

Original expression | Reduced expression (without restriction) | ||
---|---|---|---|

Evaluation at start color #7854ab, x, equals, 0, end color #7854ab | $\begin{aligned}\dfrac{(\purpleD{0})^2+3(\purpleD{0})}{(\purpleD{0})^2+5(\purpleD{0})}&=\dfrac{0}{0}\\\\&=\text{undefined}\end{aligned}$ | $\begin{aligned}\dfrac{\purpleD{0}+3}{\purpleD{0}+5}&=\dfrac{3}{5}\\\\&\phantom{\text{undefined}}\end{aligned}$ |

Because the two expressions must be equivalent for

*all*possible inputs, we must require x, does not equal, 0 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!

This becomes clear when looking at a numerical example. For example, suppose start color #7854ab, x, equals, 2, end color #7854ab.

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

## Example 2: Reducing start fraction, x, squared, minus, 9, divided by, x, squared, plus, 5, x, plus, 6, end fraction to lowest terms

**Step 1: Factor the numerator and denominator**

**Step 2: List restricted values**

Since division by 0 is undefined, here we see that start color #0c7f99, x, does not equal, minus, 2, end color #0c7f99 and start color #7854ab, x, does not equal, minus, 3, end color #7854ab.

**Step 3: Cancel common factors**

Notice that the numerator and denominator share a common factor of start color #208170, x, plus, 3, end color #208170. This can be canceled out.

**Step 4: Final answer**

We write the reduced form as follows:

start fraction, x, minus, 3, divided by, x, plus, 2, end fraction for x, does not equal, minus, 3

The original expression requires x, does not equal, minus, 2, comma, minus, 3. We do not need to note that x, does not equal, minus, 2, since this is understood from the expression.

### Check for understanding

## Want to join the conversation?

- I don't understand much the restricted values. Can someone help me to understand it please?(5 votes)
- If we have an equation (x^2 + 3x) / (x^2 + 5x) , we can simplify it to (x+3) / (x+5) . At first glance, the two equations seem to be equal, but they are actually not!

Remember that x can be any value - let just try the values -1, 1 and 0 for x in the two equations above.

For x = -1,

(x^2 + 3x) / (x^2 + 5x) = (1 - 3) / (1 - 5) = -2/-4 = 0.5

(x+3) / (x+5) = (-1 + 3)/ (-1 + 5) = 2/4 = 0.5

Both equations are equivalent as both of them give the value 0.5

For x = 1,

(x^2 + 3x) / (x^2 + 5x) = (1 + 3) / (1 + 5) = 4/6 = 2/3

(x+3) / (x+5) = (1 + 3)/ (1 + 5) = 4/6 = 2/3

Both equations are equivalent as both of them give the value 2/3

For x = 0,

(x^2 + 3x) / (x^2 + 5x) = (0 + 0) / (0 + 0) = undefined !

(x+3) / (x+5) = (0 + 3)/ (0 + 5) = 3/5 !

Both equations are NOT equivalent as both of them give different values in the case of x =0

So from the above, whenever the denominator results in a value of 0, we get an undefined value for the expression. Therefore we need to put the restriction x not equal to 0 in this case on the equation (x+3) / (x+5). With this restriction, the simplified equation is now equivalent to the original rational expression.(21 votes)

- How do I remember all of these steps?(3 votes)
- Practice, practice, practice. Eventually, you will do it enough that you don’t even think about it. You could come up with an acronym to help you, but the final goal is that you won’t need it.(12 votes)

- In #3, why can't x be equal to 1? I understand why it can't equal -1, but I don't see where 1 causes the expression to be undefined.(5 votes)
- Factor both the numerator and the denominator.

Numerator = (x-2)(x-1)

Denominator = (x-1)(x+1)

The factor of (x-1) in the denominator would cause the an undefined if x=1

Even though this factor cancels out, you need to maintain the restriction that x can't = 1 to be consistent with the original expression.

Hope this helps.(7 votes)

- This is very confusing? How will this help us in real life?(2 votes)
- It probably, most likely, won't. although some parts of the business and trade use this math, although I don't know any, there might be some.(10 votes)

- what makes an expression undefined?(3 votes)
- With rational expressions (fractions), they become undefined if the denominator = 0. If a denominator = 0, then we are dividing by 0, which is undefined.(6 votes)

- Just to get this clear, the way you figure out restricted values is like this, right?:

Let's say you have and equation that you simplified to q+1/q+2.

The restriction is that q cannot be equal to -2, correct?

I'm studying for SSATs, so I need to make sure I know this stuff. It's always been a little confusing, but now I think I've got it. Just need this clarified. Also, when in the article it said that "We do not need to note that x≠−5, since this is understood from the expression." Why don't we need to note that? How is it understood? Why wouldn't you list all the restrictions, instead of just one?(3 votes)- Let's use the expression:

(x+1)(x+2) divided by (x+3)(x+2)

This expression is undefined at x=-3 and x=-2 because both values will make the denominator equal to 0. Division by 0 is undefined. You can factor out (x+2) from both the numerator and the denominator, resulting in (x+1) divided by (x+3).

Try plugging in x=-2 into both expressions. Even though the original expression was undefined for x=-2, the simplified expression gets a different answer (-1) for that same input. Therefore, the expressions are not truly equal unless we add the restriction that x cannot equal -2.

Now try x=-3. Notice that both expressions are undefined for x=-3. Therefore, the expressions are still equivalent. You can also say that the simplified expression "implies" that x=-3 is undefined because the simplified expression is undefined for x=-3. Therefore it isn't required to remind everyone that x cannot equal -3 because you can figure that out with just the simplified expression. You can state as such if you want. On the other hand, you have to state x cannot equal -2 because that is not "implied" in the simplified expression.

I hope that helped. I know it was a lot to take in...(4 votes)

- hmmm it's a bit weird, but ok(4 votes)
- I understand that the denominator can not be 0 but i do not understand how we can tell what the numerator should be. Take example 3 I understood what x is not equal to -1 but I do not understand why x can not be 1 please explain?(3 votes)
- In the original equation, the denominator is ((x^2) +1 If x=1, then ((1^2) + 1) simplifies to (1-1) and to 0. The denominator cannot be zero, so it would not work. Hope I helped :)(0 votes)

- In Example 1 if x= -5 at any step this would make the whole expression undefined giving a zero in the denominator. Why is only x can not equal 0 listed and not x can not equal -5?(0 votes)
- You are correct. But Sal is concerned about the now missing (after x/x is eliminated) x in the denominator, and not (x+5), which is still there to see.(2 votes)

- Shouldn't step 4 of example 2 the numbers that are not answers should be -2 and 3 not -2 and -3?(0 votes)
- No, because the factor that was divided out of numerator and denominator was ( x + 3 ), and the value of x that makes that zero is -3.(6 votes)