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Section 1.1 : Review : Functions

13. Determine all the roots of \(g\left( x \right) = {x^3} + 7{x^2} - x\).

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Set the equation equal to zero and factor the left side as much as possible.

\[{x^3} + 7{x^2} - x = x\left( {{x^2} + 7x - 1} \right) = 0\]

So, we can see that one root is \(x = 0\) and because the quadratic doesn’t factor we’ll need to use the quadratic formula on that to get the remaining two roots.

\[x = \frac{{ - 7 \pm \sqrt {{{\left( 7 \right)}^2} - 4\left( 1 \right)\left( { - 1} \right)} }}{{2\left( 1 \right)}} = \frac{{ - 7 \pm \sqrt {53} }}{2}\]

We then have the following three roots of the function,

\[x = 0,\,\,\,\,\,\,\,\frac{{ - 7 + \sqrt {53} }}{2} = 0.140055,\,\,\,\,\,\,\,\frac{{ - 7 - \sqrt {53} }}{2} = - 7.140055\]