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### Course: Integrated math 3 > Unit 13

Lesson 1: Cancelling common factors- Reducing rational expressions to lowest terms
- Intro to rational expressions
- 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)

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# Simplifying rational expressions (advanced)

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 $\text{}{\displaystyle \frac{10{x}^{3}}{2{x}^{2}-18x}}$

**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.

**Step 2: List restricted values**

From the factored form, we see thatand $x\ne 0$ . $x\ne 9$

**Step 3: Cancel common factors**

**Step 4: Final answer**

We write the simplified form as follows:

for $\frac{5{x}^{2}}{x-9}$ $x\ne 0$

### 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

## Example 2: Simplifying $\text{}{\displaystyle \frac{(3-x)(x-1)}{(x-3)(x+1)}}$

**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 $x-3$ .

**Step 2: List restricted values**

From the factored form, we see that $x\ne 3$ and $x\ne -1$ .

**Step 3: Cancel common factors**

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:

### Main takeaway

The factors $x-3$ and $3-x$ are $-1\cdot (x-3)=3-x$ .

**opposites**sinceIn this example, we saw that these factors canceled, but that a factor of $-1$ was added. In other words, the factors $x-3$ and $3-x$ $\mathit{\text{-1}}$ .

*canceled to*In general opposite factors $a-b$ and $b-a$ will cancel to $-1$ provided that $a\ne b$ .

### Check your understanding

## Let's try some more problems

## Want to join the conversation?

- I'm completely lost on Rational expressions. I know how to factor, however I don't understand the whole restriction thing.(8 votes)
- 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)

- 3x+1/x : express as rational expression(6 votes)
- When we multiply the 3x term by x/x to get a common denominator, we get: (3x^2 + 1)/x(7 votes)

- 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)- Your answer is correct. For some reason, KA is setup so it doesn't except some small variations in correct answers. You may want to report it as a problem within the exercise so maybe they'll change it.(5 votes)

- it is pretty hard and specially the one where you have to choose 3 answers(4 votes)
- I totally get it(1 vote)

- How do you simplify (1/x)/1/2rootx(2 votes)
*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)

- 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)
- You're right. Since 4x^2 is in the denominator of the answer, I didn't think to include x != 0. And neither did KA, since it's still in the answer, so "everybody knows".(2 votes)

- (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)- Hint: Multiply out: (x-3)(x+6) to find the value of "c"(3 votes)

- why cant we just skip or get the answer!(2 votes)
- 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) - shouldn't (3-2x) and (2x-3) simplify to (+1) when divided, not (-1) because they are both negative already?(1 vote)
- 3-2x and 2x-3 are not both negative. If one is negative, then the other is positive, since 2x-3 is -1·(3-2x). At best, they may both be 0 (if x=3/2), but then we can't divide them, since we can't divide by 0.(3 votes)