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# 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, start fraction, x, squared, divided by, x, plus, 3, end fraction 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, start fraction, x, squared, divided by, x, plus, 3, end fraction represents the division of x, squared by x, plus, 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 start fraction, 5, divided by, 10, end fraction; we can factor out a 5 from both the numerator and the denominator and cancel them, leaving us with start fraction, 1, divided by, 2, end fraction.
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 start fraction, x, plus, 1, divided by, 2, x, plus, 2, end fraction by factoring out x, plus, 1 from both the numerator and the denominator and canceling them, leaving us with start fraction, 1, divided by, 2, end fraction.
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
start fraction, x, squared, minus, 5, x, plus, 6, divided by, x, squared, plus, x, minus, 6, end fraction, equals, start fraction, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, minus, 3, right parenthesis, divided by, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, plus, 3, right parenthesis, end fraction
The rational expression start fraction, x, squared, minus, 5, x, plus, 6, divided by, x, squared, plus, x, minus, 6, end fraction 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.
start fraction, x, squared, minus, 5, x, plus, 6, divided by, x, squared, plus, x, minus, 6, end fraction 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:
start fraction, start color #7854ab, a, end color #7854ab, divided by, start color #ca337c, b, end color #ca337c, end fraction, dot, start fraction, start color #208170, c, end color #208170, divided by, start color #a75a05, d, end color #a75a05, end fraction, equals, start fraction, start color #7854ab, a, end color #7854ab, start color #208170, c, end color #208170, divided by, start color #ca337c, b, end color #ca337c, start color #a75a05, d, end color #a75a05, end fraction
When dividing two expressions, recall that dividing by a fraction is equivalent to multiplying by that fraction's reciprocal:
start fraction, left parenthesis, start fraction, start color #7854ab, a, end color #7854ab, divided by, start color #ca337c, b, end color #ca337c, end fraction, right parenthesis, divided by, left parenthesis, start fraction, start color #208170, c, end color #208170, divided by, start color #a75a05, d, end color #a75a05, end fraction, right parenthesis, end fraction, equals, start fraction, start color #7854ab, a, end color #7854ab, divided by, start color #ca337c, b, end color #ca337c, end fraction, dot, start fraction, start color #a75a05, d, end color #a75a05, divided by, start color #208170, c, end color #208170, end fraction, equals, start fraction, start color #7854ab, a, end color #7854ab, start color #a75a05, d, end color #a75a05, divided by, start color #ca337c, b, end color #ca337c, start color #208170, c, end color #208170, end fraction
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 start fraction, x, squared, divided by, x, plus, 3, end fraction and start fraction, x, plus, 3, divided by, x, end fraction ?

If f, left parenthesis, x, right parenthesis, equals, start fraction, x, squared, plus, 2, x, plus, 1, divided by, x, plus, 3, end fraction and g, left parenthesis, x, right parenthesis, equals, start fraction, x, divided by, x, squared, plus, 4, x, plus, 3, end fraction, what is start fraction, f, left parenthesis, x, right parenthesis, divided by, g, left parenthesis, x, right parenthesis, end fraction ?

### Try it!

try: identify factorable numerators and denominators
start fraction, x, squared, plus, 1, divided by, x, plus, 1, end fraction, dot, start fraction, x, squared, plus, 3, x, plus, 2, divided by, 3, x, squared, plus, 3, end fraction
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:
start fraction, start color #7854ab, a, end color #7854ab, divided by, start color #ca337c, b, end color #ca337c, end fraction, plus minus, start fraction, start color #208170, c, end color #208170, divided by, start color #a75a05, d, end color #a75a05, end fraction, equals, start fraction, start color #7854ab, a, end color #7854ab, start color #a75a05, d, end color #a75a05, divided by, start color #ca337c, b, end color #ca337c, start color #a75a05, d, end color #a75a05, end fraction, plus minus, start fraction, start color #ca337c, b, end color #ca337c, start color #208170, c, end color #208170, divided by, start color #ca337c, b, end color #ca337c, start color #a75a05, d, end color #a75a05, end fraction, equals, start fraction, start color #7854ab, a, end color #7854ab, start color #a75a05, d, end color #a75a05, plus minus, start color #ca337c, b, end color #ca337c, start color #208170, c, end color #208170, divided by, start color #ca337c, b, end color #ca337c, start color #a75a05, d, end color #a75a05, end fraction
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, start color #ca337c, b, end color #ca337c and start color #a75a05, d, end color #a75a05.
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 start fraction, x, squared, divided by, x, plus, 3, end fraction and start fraction, 4, x, plus, 3, divided by, x, plus, 3, end fraction ?

What is the difference start fraction, 2, x, divided by, x, plus, 3, end fraction, minus, start fraction, 3, divided by, x, plus, 1, end fraction ?

### Try it!

try: rewrite expressions with common denominators
Celeste wants to find the sum of start fraction, 3, divided by, x, end fraction and start fraction, x, divided by, 2, x, minus, 3, end fraction. In order to add the two rational expressions, she must first find a common denominator by
the two denominators, x and 2, x, minus, 3.
Next, she needs to rewrite each rational expression using the common denominator. The first term can be rewritten as start fraction, 3, left parenthesis, 2, x, minus, 3, divided by, x, left parenthesis, 2, x, minus, 3, right parenthesis, end fraction, 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 start fraction, a, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction, where a and b are polynomial expressions in terms of x.
For example, for a, left parenthesis, x, right parenthesis, equals, x, squared, plus, 2, x, plus, 4 and b, left parenthesis, x, right parenthesis, equals, x, plus, 3, start fraction, a, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction, equals, start fraction, x, squared, plus, 2, x, plus, 4, divided by, x, plus, 3, end fraction.
When dividing a and b, we can find quotient polynomial q and remainder polynomial r such that:
start fraction, a, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction, equals, q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction
Where the degree of r is less than the degree of b. Since b is usually a first degree polynomial (a, x, plus, 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 start fraction, x, squared, plus, 2, x, plus, 4, divided by, x, plus, 3, end fraction, the highest degree term in the numerator is x, squared, and the highest degree term in the denominator is x. The first question we ask is "what is x, squared divided by x ?"
start fraction, x, squared, divided by, start color #208170, x, end color #208170, end fraction, equals, start color #7854ab, x, end color #7854ab, so we write start color #7854ab, x, end color #7854ab as the first term of the quotient q, find the product of start color #7854ab, x, end color #7854ab and the divisor start color #208170, x, end color #208170, plus, 3, then subtract the product from a. This eliminates the x, squared term in the dividend.
\begin{aligned} &\purpleD{x} \\ \tealE{x}+3\, | \,&\overline{x^2+2x+4}& \\ -(&x^2+3x ) \\ &\overline{\phantom{x^2~~}-x+4} \end{aligned}
Next, we do the same to what's left of the dividend, minus, x, plus, 4. We ask "what is minus, x divided by x ?"
start fraction, minus, x, divided by, start color #208170, x, end color #208170, end fraction, equals, start color #7854ab, minus, 1, end color #7854ab, so we write start color #7854ab, minus, 1, end color #7854ab as the second term of the quotient q, find the product of start color #7854ab, minus, 1, end color #7854ab and the divisor start color #208170, x, end color #208170, plus, 3, then subtract the product from minus, x, plus, 4. This eliminates the x term in the dividend.
\begin{aligned} &\purpleD{x-1} \\ \tealE{x}+3\, | \,&\overline{x^2+2x+4}& \\ -(&x^2+3x) \\ &\overline{\phantom{x^2~~}-x+4} \\ -&\phantom{x^2~~}(-x-3) \\ &\overline{\phantom{x^2+2x+}\maroonD{7}} \end{aligned}
This leaves us with the constant start color #ca337c, 7, end color #ca337c. Since the degree of start color #ca337c, 7, end color #ca337c is lower than the degree of start color #208170, x, end color #208170, plus, 3, we can stop dividing here and write our quotient and remainder.
• q, left parenthesis, x, right parenthesis, equals, start color #7854ab, x, minus, 1, end color #7854ab
• r, left parenthesis, x, right parenthesis, equals, start color #ca337c, 7, end color #ca337c
Therefore, start fraction, x, squared, plus, 2, x, plus, 4, divided by, x, plus, 3, end fraction, equals, start color #7854ab, x, minus, 1, end color #7854ab, plus, start fraction, start color #ca337c, 7, end color #ca337c, divided by, x, plus, 3, end fraction.
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 parenthesis, x, right parenthesis and b, left parenthesis, x, right parenthesis 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 parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction.

Example: For f, left parenthesis, x, right parenthesis, equals, x, squared and g, left parenthesis, x, right parenthesis, equals, x, plus, 3, rewrite start fraction, f, left parenthesis, x, right parenthesis, divided by, g, left parenthesis, x, right parenthesis, end fraction in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, g, left parenthesis, x, right parenthesis, end fraction.

### Try it!

Try: take the first steps of polynomial long division
start fraction, 6, x, cubed, plus, 3, x, squared, minus, 14, x, minus, 2, divided by, 2, x, plus, 1, end fraction
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, 2, x.
The first term of the quotient is
.

Practice: multiply two rational expressions
Which of the following is equivalent to start fraction, 2, divided by, x, plus, 1, end fraction, dot, start fraction, x, plus, 1, divided by, x, plus, 2, end fraction ?

Practice: multiply two rational expressions
start fraction, x, squared, plus, 5, x, plus, 6, divided by, x, plus, 4, end fraction, dot, start fraction, x, squared, plus, 4, x, divided by, x, plus, 3, end fraction
Which of the following is equivalent to the expression above?

practice: subtract two rational expressions
start fraction, 4, divided by, x, plus, 1, end fraction, minus, start fraction, 3, divided by, 2, x, minus, 3, end fraction
Which of the following is equivalent to the expression above?