Rational expressions space fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem facility because castle contain variables, they can be simplified in the same method that numeric fractions, likewise called numerical fractions, room simplified.

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The first step in simple a reasonable expression is to recognize the domain, the collection of all possible values of the variables. The denominator in a portion cannot be zero because department by zero is undefined. The reason

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 is that when you main point the answer 2, times the divisor 3, you get back 6. To be able to divide any kind of number c through zero
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 you would have actually to find a number that when you multiply it through 0 you would certainly get back c
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. There room no number that can do this, so us say “division through zero is undefined”. In simplifying rational expressions you should pay fist to what worths of the variable(s) in the expression would certainly make the denominator same zero. These worths cannot be contained in the domain, so they\"re called excluded values. Discard them best at the start, before you go any kind of further.

(Note that although the denominator cannot be tantamount to 0, the molecule can—this is why you only look for excluded values in the denominator the a reasonable expression.)

For rational expressions, the domain will exclude worths for which the value of the denominator is 0. Two examples to show finding the domain of an expression are shown below.


Example

Problem

Identify the domain of the expression.

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x – 4 = 0

Find any values for x that would make the denominator same 0.

x = 4

When x = 4, the denominator is same to 0.

Answer

The domain is all real numbers, other than 4.


You discovered that x can not be 4. (Sometimes you might see this idea presented together “x ≠ 4.”) What happens if you do substitute that value into the expression?


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You discover that when x = 4, the numerator evaluates come 14, however the denominator evaluate to 0. And also since department by 0 is undefined, this should be an to exclude, value.

Let\"s shot one that\"s a little more challenging.


Example

Problem

Identify the domain of the expression.

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Find any kind of values because that x that would certainly make the denominator same to 0 by setting the denominator equal to 0 and also solving the equation.

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Solve the equation through factoring. The services are the values that space excluded native the domain.

Answer

The domain is all real numbers other than −9 and also 1.



Find the domain the the rational expression

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.

A) all genuine numbers except −4

B) all real numbers other than 4

C) all real numbers except 0

D) all actual numbers


Show/Hide Answer

A) all actual numbers except −4

Correct. As soon as x = −4, the denominator is 2(−4) + 8 = −8 + 8 = 0. Division by 0 is undefined, so the domain need to exclude x = −4.

B) all real numbers other than 4

Incorrect. When x = 4, the denominator does no equal 0, thus it is no an excluded value. Set the denominator equal to 0 and solve because that x. The exactly answer is all real numbers other than −4.

C) all genuine numbers other than 0

Incorrect. As soon as x = 0, the numerator amounts to 0 but the denominator does not, because of this it is not an to exclude, value. Collection the denominator equal to 0 and solve because that x. The correct answer is all actual numbers other than −4.

D) all genuine numbers

Incorrect. Over there is one value of x that will make the denominator 0. Set the denominator equal to 0 and also solve because that x. The exactly answer is all actual numbers except −4.

Simplifying reasonable Expressions


Once you\"ve established the to exclude, values, the following step is to leveling the reasonable expression. To simplify a reasonable expression, follow the same technique you use to leveling numeric fractions: find typical factors in the numerator and denominator. Let’s begin by simple a numeric fraction.


Example

Problem

Simplify.

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Factor the numerator and denominator.

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Identify fractions the equal 1, and also then traction them the end of the fraction. In this fraction, the variable 3 is in both the numerator and also denominator. Recall the

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 is another name for 1.

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Simplify.

Answer

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Now, you can have done that problem in your head—but it to be worth composing it all down, because that\"s exactly how you leveling a rational expression.

So let\"s leveling a reasonable expression, using the same method you applied to the fraction

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. Just this time, the numerator and also denominator room both monomials through variables.


Example

Problem

Simplify.

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Factor the numerator and denominator.

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Identify fractions that equal 1, and also then traction them out of the fraction.

Simplify.

Answer

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See—the same steps functioned again. Element the numerator, aspect the denominator, identify determinants that are typical to the numerator and denominator, and also write together a element of 1, and simplify.

When simplifying reasonable expressions, the is a an excellent habit to constantly consider the domain, and also to discover the worths of the change (or variables) that make the expression undefined. (This will certainly come in handy once you begin solving for variables a bit later on.)


Example

Problem

Identify the domain of the expression.

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Find any kind of values because that x that would certainly make the denominator equal to 0 by setting the denominator equal to 0 and solving the equation.

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x = 0

The values for x that make the denominator equal 0 are excluded indigenous the domain.

Answer

The domain is all genuine numbers except 0.


Notice that you began with the initial expression, and also identified values of x that would make 25x equal to 0. Why does this matter? Look in ~

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 when that is simplified…it is the portion . Due to the fact that 5 is the denominator, it appears that no values should be excluded indigenous the domain. When finding the domain of an expression, you always start with the original expression due to the fact that variable terms may be factored out as part of the simplification process.

In the examples that follow, the numerator and also the denominator are polynomials with an ext than one term, however the same principles of simple will as soon as again apply. Variable the numerator and denominator to simplify the rational expression.


Example

Problem

Simplify and state the domain because that the expression.

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x2 + 12x + 27 = 0

(x + 3)(x + 9) = 0

x + 3 = 0

or

x + 9 = 0

x = 0 – 3

or

x = 0 – 9

x = −3

or

x = −9

x = −3 or x = −9

domain is all real numbers other than −3 and −9

To find the domain (and the to exclude, values), discover the worths for which the denominator is equal to 0. Factor the quadratic to uncover the values.

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Factor the numerator and denominator.

Identify the determinants that space the same in the numerator and denominator.

Write as separate fractions, pulling the end fractions that equal 1.

Simplify.

Answer

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The domain is all genuine numbers except −3 and also −9.

Example

Problem

Simplify and state the domain for the expression.

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x3 – x2 – 20x  = 0

x(x2 – x – 20) = 0

x(x – 5)(x + 4) = 0

domain is all actual numbers other than 0, 5, and −4

To find the domain, identify the worths for i beg your pardon the denominator is equal to 0.

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To simplify, element the numerator and also denominator the the reasonable expression. Recognize the components that space the exact same in the numerator and denominator.

Write as different fractions, pulling the end fractions the equal 1.

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 or
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Simplify. The is acceptable to either leave the denominator in factored type or to distribution multiplication.

Answer

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 or
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The domain is all genuine numbers other than 0, 5, and −4.


Steps because that Simplifying a rational Expression

To simplify a reasonable expression, follow this steps:

· recognize the domain. The to exclude, values are those values for the change that an outcome in the expression having a denominator of 0.

· variable the numerator and also denominator.

· Find usual factors because that the numerator and denominator and simplify.

Simplify the rational expression below.

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A)

B)

C)

D)


Show/Hide Answer

A)

Incorrect. Friend must very first factor the polynomials in the numerator and also the denominator and then express like factors in the numerator and also denominator as 1 come simplify. The expression have the right to be factored together , therefore the exactly answer is .

B)

Correct. The reasonable expression deserve to be simplified by factoring the numerator and also denominator together . Due to the fact that

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, leveling the expression come .

C)

Incorrect. Friend must first factor the polynomials in the numerator and the denominator and also then express like determinants in the numerator and also denominator together 1 come simplify. The expression can be factored as , for this reason the exactly answer is .

D)

Incorrect.  cannot be simplified to  because the x’s in the numerator and denominator space not being multiplied, they are being included The exactly answer is .

Summary


Rational expressions room fractions containing polynomials. They have the right to be simplified much like numeric fractions. To leveling a reasonable expression, very first determine typical factors that the numerator and also denominator, and then remove them by rewriting them together expressions equal to 1.

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An extr consideration for rational expression is to determine what values are excluded indigenous the domain. Since division by 0 is undefined, any values of the variables that an outcome in a denominator that 0 need to be excluded. To exclude, values must be established in the original equation, not from that is factored form.