# 1.6 Rational expressions  (Page 2/6)

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

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 $\text{\hspace{0.17em}}\frac{ab}{1+bc}.$

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin