Rational expressions space fractions that have actually a polynomial in the numerator, denominator, or both. Although reasonable expressions deserve to seem facility because they contain variables, they have the right to be simplified in the same way that numeric fractions, also called numerical fractions, room simplified.

You are watching: Find any values for which the rational expression is undefined


The an initial step in simplifying a reasonable expression is to determine the domain, the collection of all possible values that the variables. The denominator in a portion cannot it is in zero because division by zero is undefined. The factor

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 is that when you multiply the prize 2, time the divisor 3, girlfriend get ago 6. To have the ability to divide any number c by zero
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 you would have to uncover a number that once you main point it by 0 you would get ago c
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. There are no number that can do this, so we say “division by zero is undefined”. In simplifying reasonable expressions you must pay fist to what worths of the variable(s) in the expression would make the denominator equal zero. These worths cannot be included in the domain, so they"re dubbed excluded values. Discard them appropriate at the start, before you go any further.

(Note the although the denominator cannot be indistinguishable to 0, the numerator can—this is why you just look because that excluded worths in the denominator the a rational expression.)

For rational expressions, the domain will certainly exclude values for i beg your pardon the value of the denominator is 0. Two instances to highlight finding the domain of an expression are presented below.


Example

Problem

Identify the domain the 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 equal to 0.

Answer

The domain is all actual numbers, except 4.


You found that x can not be 4. (Sometimes you might see this idea presented together “x ≠ 4.”) What wake up if you carry out substitute that value right into the expression?


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You discover that once x = 4, the molecule evaluates to 14, but the denominator evaluate to 0. And since division by 0 is undefined, this need to be an excluded value.

Let"s shot one that"s a little an ext challenging.


Example

Problem

Identify the domain the the expression.

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

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Solve the equation by factoring. The remedies are the worths that room excluded native the domain.

Answer

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



Find the domain of the reasonable expression

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.

A) all genuine numbers except −4

B) all actual numbers other than 4

C) all real numbers other than 0

D) all real numbers


Show/Hide Answer

A) all real numbers other than −4

Correct. When x = −4, the denominator is 2(−4) + 8 = −8 + 8 = 0. Department by 0 is undefined, so the domain must exclude x = −4.

B) all actual numbers except 4

Incorrect. When x = 4, the denominator does not equal 0, thus it is no an to exclude, value. Collection the denominator same to 0 and solve for x. The correct answer is all genuine numbers except −4.

C) all genuine numbers except 0

Incorrect. As soon as x = 0, the numerator amounts to 0 however the denominator walk not, as such it is no an to exclude, value. Collection the denominator same to 0 and also solve because that x. The exactly answer is all real numbers except −4.

D) all actual numbers

Incorrect. There is one value of x that will make the denominator 0. Collection the denominator same to 0 and solve because that x. The correct answer is all genuine numbers other than −4.

Simplifying rational Expressions


Once you"ve figured out the to exclude, values, the following step is to simplify the reasonable expression. To leveling a reasonable expression, follow the same strategy you use to leveling numeric fractions: find usual 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 pull them the end of the fraction. In this fraction, the aspect 3 is in both the numerator and denominator. Recall that

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 is an additional name because that 1.

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Simplify.

Answer

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Now, you could have excellent that problem in your head—but it to be worth composing it every down, due to the fact that that"s specifically how you leveling a reasonable expression.

So let"s leveling a reasonable expression, making use of the same technique you used to the fraction

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. Only this time, the numerator and also denominator space both monomials v variables.


Example

Problem

Simplify.

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Factor the numerator and also denominator.

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

Simplify.

Answer

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See—the exact same steps operated again. Element the numerator, variable the denominator, identify factors that are usual to the numerator and also denominator, and write together a factor of 1, and simplify.

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


Example

Problem

Identify the domain that the expression.

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

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x = 0

The worths for x that make the denominator equal 0 room excluded native the domain.

Answer

The domain is all real numbers other than 0.


Notice that you began with the original expression, and also identified worths of x that would make 25x equal to 0. Why go this matter? Look at

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 when the is simplified…it is the fraction . Since 5 is the denominator, it appears that no values should be excluded indigenous the domain. Once finding the domain of one expression, you always start v the initial expression due to the fact that variable terms may be factored the end as component of the leveling process.

In the instances that follow, the numerator and the denominator room polynomials with more than one term, yet the same principles of simplifying will when again apply. Variable the numerator and also denominator to leveling the reasonable expression.


Example

Problem

Simplify and also state the domain for 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 except −3 and −9

To find the domain (and the excluded values), discover the values for i beg your pardon the denominator is same to 0. Aspect the quadratic to find the values.

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Factor the numerator and also denominator.

Identify the factors that room the same in the numerator and denominator.

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

Simplify.

Answer

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The domain is all real numbers except −3 and −9.

Example

Problem

Simplify and also state the domain because that 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 also −4

To uncover the domain, identify the worths for which the denominator is equal to 0.

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To simplify, variable the numerator and also denominator of the reasonable expression. Determine the components that room the same in the numerator and also denominator.

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

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 or
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Simplify. That is acceptable to either leave the denominator in factored form or to distribution multiplication.

Answer

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 or
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The domain is all real numbers except 0, 5, and also −4.


Steps because that Simplifying a reasonable Expression

To leveling a reasonable expression, follow these steps:

· recognize the domain. The exclude, values are those worths for the change that result in the expression having a denominator that 0.

· factor the numerator and also denominator.

· Find usual factors for the numerator and also denominator and simplify.

Simplify the reasonable expression below.

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* and also −10.>

A)

B)

C)

D)


Show/Hide Answer

A)

Incorrect. Friend must an initial factor the polynomials in the numerator and also the denominator and also then refer like determinants in the numerator and denominator together 1 to simplify. The expression have the right to be factored as , so the exactly answer is .

B)

Correct. The rational expression have the right to be streamlined by factoring the numerator and denominator as . Because

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, simplify the expression come .

C)

Incorrect. Girlfriend must an initial factor the polynomials in the numerator and also the denominator and also then to express like components in the numerator and also denominator as 1 to simplify. The expression have the right to be factored together , so the exactly answer is .

D)

Incorrect.  cannot be simplified to  because the x’s in the numerator and also denominator are not gift multiplied, they space 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 simplify a reasonable expression, very first determine typical factors that the numerator and denominator, and also then eliminate them by rewriting them together expressions same to 1.

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An extr consideration because that rational expression is to recognize what values space excluded native the domain. Since department by 0 is undefined, any values the the variables that result in a denominator of 0 have to be excluded. Excluded values should be established in the initial equation, not from the factored form.