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# Algebraic expressions

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For what real values of a is the expression (a + 3)/(a^3 - a)  undefined? List your answers in increasing order separated by commas.

Jul 3, 2021

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first of all lets evaluate a bit:

$\frac{a+3}{a^3-a}$

$\frac{a+3}{a\left(a^2-1\right) }$

$\frac{a+3}{a\left(a+1\right)\left(a-1\right)}$

now, any rational expression will be undefined when the denominator is $=\: 0$ right? because dividing by 0 ( consider it like like something which does not exist ) just makes no sense. so we note that when $a\left(a+1\right)\left(a-1\right)=0$, that expression is undefined:

$a\left(a+1\right)\left(a-1\right)=0$

$\left(a+1\right)\left(a-1\right)=\frac{0}{a}$

$\left(a+1\right)\left(a-1\right)=0$       $\implies \begin{cases} a+1=0 \\ \ \ \ \ \text{and} \\ a-1=0 \end{cases} \implies \begin{cases} a=-1 \\ \ \ \text{and} \\ a=1 \end{cases}$

thus, whenever $\boxed{a=1,\: -1}$ the fraction becomes undefined

Jul 3, 2021