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### Course: Digital SAT Math>Unit 4

Lesson 4: Operations with rational expressions: foundations

# Operations with rational expressions | Lesson

A guide to operations with rational expressions on the digital SAT

## What are rational expressions?

Rational expressions look like fractions that have variables in their denominators (and often numerators too). For example, $\frac{{x}^{2}}{x+3}$ is a rational expression. Just as we can add, subtract, multiply, and divide fractions, we can perform the four operations on rational expressions.
Rational expressions also represent the division of one polynomial expression by another. For example, $\frac{{x}^{2}}{x+3}$ represents the division of ${x}^{2}$ by $x+3$, for which we can find a quotient and a remainder.
In this lesson, we'll learn to:
1. Simplify rational expressions
2. Add, subtract, multiply, and divide rational expressions
3. Rewrite rational expressions in the form of quotients and remainders
This lesson builds upon the following skills:
• Factoring quadratic and polynomial expressions
• Operations with polynomials
You can learn anything. Let's do this!

## How do I simplify rational expressions?

### Intro to rational expression simplification

Reducing rational expressions to lowest termsSee video transcript

### Just like fractions, but with polynomial factoring

You're probably familiar with simplifying fractions like $\frac{5}{10}$; we can factor out a $5$ from both the numerator and the denominator and cancel them, leaving us with $\frac{1}{2}$.
With rational expressions, we can also cancel out factors that appear in both the numerator and the denominator. These factors can be polynomials!
For example, we can simplify the rational expression $\frac{x+1}{2x+2}$ by factoring out $x+1$ from both the numerator and the denominator and canceling them, leaving us with $\frac{1}{2}$.
On the SAT, the numerators and denominators of rational expressions can also be quadratic expressions and higher order polynomials, so the ability to factor these expressions fluently is key to your success.

### Try it!

try: find the simplified rational expression
$\frac{{x}^{2}-5x+6}{{x}^{2}+x-6}=\frac{\left(x-2\right)\left(x-3\right)}{\left(x-2\right)\left(x+3\right)}$
The rational expression $\frac{{x}^{2}-5x+6}{{x}^{2}+x-6}$ is factored as shown above. Since both the numerator and the denominator contain
as a factor, we can cancel the identical factors to simplify the expression.
$\frac{{x}^{2}-5x+6}{{x}^{2}+x-6}$ is equivalent to
.

## How do I multiply and divide rational expressions?

### Multiplying & dividing rational expressions: monomials

Multiplying & dividing rational expressions: monomialsSee video transcript

### Multiplying and dividing rational expressions

The same rules for multiplying and dividing fractions apply to multiplying and dividing rational expressions.
When multiplying two rational expressions:
$\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$
When dividing two expressions, recall that dividing by a fraction is equivalent to multiplying by that fraction's reciprocal:
$\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{d}\right)}=\frac{a}{b}\cdot \frac{d}{c}=\frac{ad}{bc}$
However, to avoid lengthy polynomial operations, it's recommended that you factor and cancel any cancellable factors before you write out the final expression.
To multiply two rational expressions:
1. Factor any factorable polynomial expressions in the numerators and the denominators.
2. Cancel any identical factors that appear in both the numerators and the denominators of the expressions.
3. Multiply the remaining numerators and multiply the remaining denominators.
Dividing two rational expressions is similar to multiplying; just remember that dividing by an expression is equivalent to multiplying by the reciprocal of the same expression.

#### Let's look at some examples!

What is the product of $\frac{{x}^{2}}{x+3}$ and $\frac{x+3}{x}$ ?
If $f\left(x\right)=\frac{{x}^{2}+2x+1}{x+3}$ and $g\left(x\right)=\frac{x}{{x}^{2}+4x+3}$, what is $\frac{f\left(x\right)}{g\left(x\right)}$ ?

### Try it!

try: identify factorable numerators and denominators
$\frac{{x}^{2}+1}{x+1}\cdot \frac{{x}^{2}+3x+2}{3{x}^{2}+3}$
Before multiplying the two rational expressions shown above, Raj wants to factor the numerators and denominators. Which of the following can be factored?

## How do I add and subtract rational expressions?

### Adding and subtracting rational expressions

The same rules for adding and subtracting fractions apply to adding and subtracting rational expressions.
When adding or subtracting two rational expressions with unlike denominators:
$\frac{a}{b}±\frac{c}{d}=\frac{ad}{bd}±\frac{bc}{bd}=\frac{ad±bc}{bd}$
Remember that you can only add and subtract the numerators of the rational expressions if the expressions have a common denominator! In most cases, the easiest way to find a common denominator is to multiply the two unlike denominators, $b$ and $d$.
To add and subtract two rational expressions:
1. Find a common denominator for the two expressions. In most cases, the product of the two denominators would work.
2. Rewrite the equivalent form of each rational expression using the common denominator.
3. Add or subtract the numerators of the expressions while retaining the common denominator.
4. Combine like terms and write the result.
5. Factor and/or cancel as needed.

#### Let's look at some examples!

What is the sum of $\frac{{x}^{2}}{x+3}$ and $\frac{4x+3}{x+3}$ ?
What is the difference $\frac{2x}{x+3}-\frac{3}{x+1}$ ?

### Try it!

try: rewrite expressions with common denominators
Celeste wants to find the sum of $\frac{3}{x}$ and $\frac{x}{2x-3}$. In order to add the two rational expressions, she must first find a common denominator by
the two denominators, $x$ and $2x-3$.
Next, she needs to rewrite each rational expression using the common denominator. The first term can be rewritten as $\frac{3\left(2x-3}{x\left(2x-3\right)}$, and the second term can be rewritten as
.

## How do I rewrite a rational expression as a quotient and a remainder?

### Dividing polynomials by linear expressions

Dividing polynomials by linear expressionsSee video transcript

### Polynomial long division

We can represent any rational expression as $\frac{a\left(x\right)}{b\left(x\right)}$, where $a$ and $b$ are polynomial expressions in terms of $x$.
For example, for $a\left(x\right)={x}^{2}+2x+4$ and $b\left(x\right)=x+3$, $\frac{a\left(x\right)}{b\left(x\right)}=\frac{{x}^{2}+2x+4}{x+3}$.
When dividing $a$ and $b$, we can find quotient polynomial $q$ and remainder polynomial $r$ such that:
$\frac{a\left(x\right)}{b\left(x\right)}=q\left(x\right)+\frac{r\left(x\right)}{b\left(x\right)}$
Where the degree of $r$ is less than the degree of $b$. Since $b$ is usually a first degree polynomial ($ax+b$) on the SAT, $r$ is usually a constant.
When dividing two polynomials using long division, we focus on the highest degree terms in the numerator and the denominator first. For example, for $\frac{{x}^{2}+2x+4}{x+3}$, the highest degree term in the numerator is ${x}^{2}$, and the highest degree term in the denominator is $x$. The first question we ask is "what is ${x}^{2}$ divided by $x$ ?"
$\frac{{x}^{2}}{x}=x$, so we write $x$ as the first term of the quotient $q$, find the product of $x$ and the divisor $x+3$, then subtract the product from $a$. This eliminates the ${x}^{2}$ term in the dividend.
Next, we do the same to what's left of the dividend, $-x+4$. We ask "what is $-x$ divided by $x$ ?"
$\frac{-x}{x}=-1$, so we write $-1$ as the second term of the quotient $q$, find the product of $-1$ and the divisor $x+3$, then subtract the product from $-x+4$. This eliminates the $x$ term in the dividend.
This leaves us with the constant $7$. Since the degree of $7$ is lower than the degree of $x+3$, we can stop dividing here and write our quotient and remainder.
• $q\left(x\right)=x-1$
• $r\left(x\right)=7$
Therefore, $\frac{{x}^{2}+2x+4}{x+3}=x-1+\frac{7}{x+3}$.
Another strategy to find the quotient and the remainder is to group the numerator, which requires us to split the numerator of a rational expression into a polynomial divisible by the denominator and the remainder.
You don't need to know how to group the numerator, but it may save you time on the test.
To divide polynomial expressions $a\left(x\right)$ and $b\left(x\right)$ using long division:
1. Divide the highest degree term of $a$ by the highest degree term of $b$. This gives you a term of the quotient.
2. Multiply the result of Step 1 by $b$.
3. Subtract the result of Step 2 from $a$. Be careful when subtracting negatives!
4. Repeat the divide-multiply-subtract steps using what's left of the dividend until the result is of a lower degree than $b$.
5. The terms calculated in the "divide" steps form the quotient $q$. The leftover polynomial with a lower degree than $b$ is the remainder $r$.
6. Write the result as $q\left(x\right)+\frac{r\left(x\right)}{b\left(x\right)}$.
Example: For $f\left(x\right)={x}^{2}$ and $g\left(x\right)=x+3$, rewrite $\frac{f\left(x\right)}{g\left(x\right)}$ in the form $q\left(x\right)+\frac{r\left(x\right)}{g\left(x\right)}$.

### Try it!

Try: take the first steps of polynomial long division
$\frac{6{x}^{3}+3{x}^{2}-14x-2}{2x+1}$
In the rational expression above, the numerator is a
polynomial and the denominator is a first degree polynomial. Therefore, the quotient must be a
polynomial.
To find the first term of the quotient, we must divide the highest degree term in the numerator,
, by the highest degree term in the denominator, $2x$.
The first term of the quotient is
.

Practice: multiply two rational expressions
Which of the following is equivalent to $\frac{2}{x+1}\cdot \frac{x+1}{x+2}$ ?

Practice: multiply two rational expressions
$\frac{{x}^{2}+5x+6}{x+4}\cdot \frac{{x}^{2}+4x}{x+3}$
Which of the following is equivalent to the expression above?

practice: subtract two rational expressions
$\frac{4}{x+1}-\frac{3}{2x-3}$
Which of the following is equivalent to the expression above?

practice: rewrite a rational expression as quotient and remainder
Which of the following is equivalent to $\frac{{x}^{2}+5x+2}{x+5}$ ?

## Want to join the conversation?

• Ugh i hope i’ll get a good grade after all of this
My exam is in the 6th of May. Pray for me!!
• Same!
• I am not getting a 1600
• you are bro i trust you
• I literally don't understand anything in this lecture... I feel really stupid...
• its okay to not understand from the very frst time.dont give up and revise the lecture for 1 2 times im sure you wiill accopmlish it !
• the last video isn't understandable
• As a student, I can confirm that people walk up to me on the steet and ask random math problems like;
quick, Divide 3x^3+4x^2-3x+7 by x+2.

And I run faster...
• i dont like this section:(