# 1.6 Rational expressions  (Page 2/6)

 Page 2 / 6
$\frac{1}{x}\cdot \frac{3}{{x}^{2}}=\frac{3}{{x}^{3}}$

Given two rational expressions, divide them.

1. Rewrite as the first rational expression multiplied by the reciprocal of the second.
2. Factor the numerators and denominators.
3. Multiply the numerators.
4. Multiply the denominators.
5. Simplify.

## Dividing rational expressions

Divide the rational expressions and express the quotient in simplest form:

$\frac{2{x}^{2}+x-6}{{x}^{2}-1}÷\frac{{x}^{2}-4}{{x}^{2}+2x+1}$
$\frac{9{x}^{2}-16}{3{x}^{2}+17x-28}÷\frac{3{x}^{2}-2x-8}{{x}^{2}+5x-14}$

Divide the rational expressions and express the quotient in simplest form:

$\frac{9{x}^{2}-16}{3{x}^{2}+17x-28}÷\frac{3{x}^{2}-2x-8}{{x}^{2}+5x-14}$

$1$

## Adding and subtracting rational expressions

Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.

$\begin{array}{ccc}\hfill \frac{5}{24}+\frac{1}{40}& =& \frac{25}{120}+\frac{3}{120}\hfill \\ & =& \frac{28}{120}\hfill \\ & =& \frac{7}{30}\hfill \end{array}$

We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.

The easiest common denominator to use will be the least common denominator    , or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were $\text{\hspace{0.17em}}\left(x+3\right)\left(x+4\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(x+4\right)\left(x+5\right),$ then the LCD would be $\text{\hspace{0.17em}}\left(x+3\right)\left(x+4\right)\left(x+5\right).$

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of $\text{\hspace{0.17em}}\left(x+3\right)\left(x+4\right)\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}\frac{x+5}{x+5}\text{\hspace{0.17em}}$ and the expression with a denominator of $\text{\hspace{0.17em}}\left(x+4\right)\left(x+5\right)\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}\frac{x+3}{x+3}.$

Given two rational expressions, add or subtract them.

1. Factor the numerator and denominator.
2. Find the LCD of the expressions.
3. Multiply the expressions by a form of 1 that changes the denominators to the LCD.
4. Add or subtract the numerators.
5. Simplify.

$\frac{5}{x}+\frac{6}{y}$

First, we have to find the LCD. In this case, the LCD will be $\text{\hspace{0.17em}}xy.\text{\hspace{0.17em}}$ We then multiply each expression by the appropriate form of 1 to obtain $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ as the denominator for each fraction.

$\begin{array}{l}\frac{5}{x}\cdot \frac{y}{y}+\frac{6}{y}\cdot \frac{x}{x}\\ \frac{5y}{xy}+\frac{6x}{xy}\end{array}$

Now that the expressions have the same denominator, we simply add the numerators to find the sum.

$\frac{6x+5y}{xy}$

## Subtracting rational expressions

Subtract the rational expressions:

$\frac{6}{{x}^{2}+4x+4}-\frac{2}{{x}^{2}-4}$

Do we have to use the LCD to add or subtract rational expressions?

No. Any common denominator will work, but it is easiest to use the LCD.

Subtract the rational expressions: $\text{\hspace{0.17em}}\frac{3}{x+5}-\frac{1}{x-3}.$

$\frac{2\left(x-7\right)}{\left(x+5\right)\left(x-3\right)}$

## Simplifying complex rational expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression $\text{\hspace{0.17em}}\frac{a}{\frac{1}{b}+c}\text{\hspace{0.17em}}$ can be simplified by rewriting the numerator as the fraction $\text{\hspace{0.17em}}\frac{a}{1}\text{\hspace{0.17em}}$ and combining the expressions in the denominator as $\text{\hspace{0.17em}}\frac{1+bc}{b}.\text{\hspace{0.17em}}$ We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get $\text{\hspace{0.17em}}\frac{a}{1}\cdot \frac{b}{1+bc},$ which is equal to $\text{\hspace{0.17em}}\frac{ab}{1+bc}.$

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Miranda
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Miranda Drice
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Miranda
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
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Propessor
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Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
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algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
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and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
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(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
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Kishu
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what is the function of sine with respect of cosine , graphically
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Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function