Exercise 2 Page 380

We are asked to find the rational expression that does not belong. To do this, we will try and simplify the given rational expressions. Let's look at the

first

expression.

\frac{x - 4}{x ^{2}}

We see that neither the numerator or denominator can be further simplified to cancel out any common factors. Therefore, this fraction is irreducible. Let's look at the

second

fraction.

\frac{x ^{2} + 4 x - 1 2}{x ^{2} + 6 x}

Let's try and simplify this.

\frac{x ^{2} + 4 x - 1 2}{x ^{2} + 6 x}

▼

Split into factors

WriteSum

Write as a sum

\frac{x ^{2} - 2 x + 6 x - 1 2}{x ^{2} + 6 x}

FactorOut

Factor out $x$

\frac{x ( x - 2 ) + ( 6 x - 1 2 )}{x ^{2} + 6 x}

FactorOut

Factor out $6$

\frac{x ( x - 2 ) + 6 ( x - 2 )}{x ^{2} + 6 x}

FactorOut

Factor out $(x - 2)$

\frac{( x - 2 ) ( x + 6 )}{x ^{2} + 6 x}

FactorOut

Factor out $x$

\frac{( x - 2 ) ( x + 6 )}{x ( x + 6 )}

CancelCommonFac

Cancel out common factors

\frac{( x - 2 ) ( x + 6 )}{x ( x + 6 )}

SimpQuot

Simplify quotient

\frac{x - 2}{x}

Let's now look at the

third

expression.

\frac{9 + x}{3 x ^{2}}

We see that neither the numerator or denominator can be further simplified to cancel out any common factors. Therefore, this fraction is irreducible. Let's look at the

fourth

fraction.

\frac{x ^{2} - x - 1 2}{x ^{2} - 6 x}

Let's try and simplify this.

\frac{x ^{2} - x - 1 2}{x ^{2} - 6 x}

▼

Split into factors

WriteDiff

Write as a difference

\frac{x ^{2} + 3 x - 4 x - 1 2}{x ^{2} - 6 x}

FactorOut

Factor out $x$

\frac{x ( x + 3 ) - 4 x - 1 2}{x ^{2} - 6 x}

FactorOut

Factor out $- 4$

\frac{x ( x + 3 ) - 4 ( x + 3 )}{x ^{2} - 6 x}

FactorOut

Factor out $(x + 3)$

\frac{( x + 3 ) ( x - 4 )}{x ^{2} - 6 x}

FactorOut

Factor out $x$

\frac{( x + 3 ) ( x - 4 )}{x ( x - 6 )}

We see that there are no common factors in the numerator and denominator that can be canceled out. Therefore, this fraction is also irreducible. The only fraction which is not irreducible is the second one,

\frac{x ^{2} + 4 x - 1 2}{x ^{2} + 6 x} .

This is the rational expression that does not belong with the other three.

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