Here is a list of all possible rational zeroes for the polynomial.

\[\begin{align*}\frac{{ \pm 1}}{{ \pm 1}} & = \pm 1 & \hspace{0.25in}\frac{{ \pm 2}}{{ \pm 1}} & = \pm 2 & \hspace{0.25in}\frac{{ \pm 3}}{{ \pm 1}} & = \pm 3 & \hspace{0.25in}\frac{{ \pm 4}}{{ \pm 1}} & = \pm 4 & \hspace{0.25in}\frac{{ \pm 6}}{{ \pm 1}} & = \pm 6 & \hspace{0.25in}\frac{{ \pm 8}}{{ \pm 1}} & = \pm 8\\ \\ & & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in}\,\,\,\,\frac{{ \pm 12}}{{ \pm 1}} & = \pm 12 & \hspace{0.25in}\,\,\,\,\,\frac{{ \pm 24}}{{ \pm 1}} & = \pm 24\\ \\ \frac{{ \pm 1}}{{ \pm 2}} & = \frac{{ \pm 1}}{{ \pm 2}} & \hspace{0.25in}\frac{{ \pm 2}}{{ \pm 2}} & = \pm 1 & \hspace{0.25in}\frac{{ \pm 3}}{{ \pm 2}} & = \frac{{ \pm 3}}{{ \pm 2}} & \hspace{0.25in}\frac{{ \pm 4}}{{ \pm 2}} & = \pm 2 & \hspace{0.25in}\frac{{ \pm 6}}{{ \pm 2}} & = \pm 3 & \hspace{0.25in}\frac{{ \pm 8}}{{ \pm 2}} & = \pm 4\\ \\ & & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in} & \hspace{0.25in}\,\,\,\,\frac{{ \pm 12}}{{ \pm 2}} & = \pm 6 & \hspace{0.25in}\,\,\,\,\,\frac{{ \pm 24}}{{ \pm 2}} & = \pm 12\end{align*}\]

So, we have a total of 20 possible zeroes for the polynomial.

Show Step 3

We now need to start the synthetic division work. We’ll start with the “small” integers first.

\[\begin{array}{r|rrrrl} {} & 2 & -7 & -2 & 28 & -24 \\ \hline -1 & 2 & -9 & 7 & 21 & -45=A\left( -1 \right)\ne 0 \\ 1 & 2 & -5 & -7 & 21 & -3=A\left( 1 \right)\ne 0 \\ -2 & 2 & -11 & 20 & -12 & 0 =A\left( -2 \right)=0!! \\ \end{array}\]

Okay we now know that \(x = - 2\) is a zero and we can write the polynomial as,

\[A\left( x \right) = 2{x^4} - 7{x^3} - 2{x^2} + 28x - 24 = \left( {x + 2} \right)\left( {2{x^3} - 11{x^2} + 20x - 12} \right)\] Show Step 4

So, now we need to continue the process using \(Q\left( x \right) = 2{x^3} - 11{x^2} + 20x - 12\). Here is a list of all possible zeroes of \(Q\left( x \right)\).

\[\begin{align*}\frac{{ \pm 1}}{{ \pm 1}} & = \pm 1 & \hspace{0.25in}\frac{{ \pm 2}}{{ \pm 1}} & = \pm 2 & \hspace{0.25in}\frac{{ \pm 3}}{{ \pm 1}} & = \pm 3 & \hspace{0.25in}\frac{{ \pm 4}}{{ \pm 1}} & = \pm 4 & \hspace{0.25in}\frac{{ \pm 6}}{{ \pm 1}} & = \pm 6 & \hspace{0.25in}\frac{{ \pm 12}}{{ \pm 1}} & = \pm 12\\ \\ \frac{{ \pm 1}}{{ \pm 2}} & = \frac{{ \pm 1}}{{ \pm 2}} & \hspace{0.25in}\frac{{ \pm 2}}{{ \pm 2}} & = \pm 1 & \hspace{0.25in}\frac{{ \pm 3}}{{ \pm 2}} & = \frac{{ \pm 3}}{{ \pm 2}} & \hspace{0.25in}\frac{{ \pm 4}}{{ \pm 2}} & = \pm 2 & \hspace{0.25in}\frac{{ \pm 6}}{{ \pm 2}} & = \pm 3 & \hspace{0.25in}\frac{{ \pm 12}}{{ \pm 2}} & = \pm 6\end{align*}\]

So we have a list of 16 possible zeroes, but note that we’ve already proved that \( \pm 1\) can’t be zeroes of the original polynomial and so can’t be zeroes of \(Q\left( x \right)\) either.

Here’s the synthetic division work for this \(Q\left( x \right)\).

\[\,\,\begin{array}{r|rrrl} {} & 2 & -11 & 20 & -12 \\ \hline -2 & 2 & -15 & 50 & -112 =Q\left( -2 \right)\ne 0 \\ 2 & 2 & -7 & 6 & 0 =Q\left( 2 \right)=0!! \\ \end{array}\]

Therefore, \(x = 2\) is a zero of \(Q\left( x \right)\) and the factored form of \(Q\left( x \right)\) is,

\[Q\left( x \right) = 2{x^3} - 11{x^2} + 20x - 12 = \left( {x - 2} \right)\left( {2{x^2} - 7x + 6} \right)\]

This also means that the factored form of the original polynomial is now,

\[A\left( x \right) = 2{x^4} - 7{x^3} - 2{x^2} + 28x - 24 = \left( {x + 2} \right)\left( {x - 2} \right)\left( {2{x^2} - 7x + 6} \right)\] Show Step 5

We’re down to a quadratic polynomial and so we can and we can just factor this to get the fully factored form of the original polynomial. This is,

\[A\left( x \right) = 2{x^4} - 7{x^3} - 2{x^2} + 28x - 24 = \left( {x + 2} \right)\left( {x - 2} \right)\left( {x - 2} \right)\left( {2x - 3} \right)=\left( {x + 2} \right){\left( {x - 2} \right)^2}\left( {2x - 3} \right)\] Show Step 6

From the fully factored form we get the following set of zeroes for the original polynomial.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = - 2\hspace{0.25in}\hspace{0.25in}x = \frac{3}{2}\hspace{0.25in}\hspace{0.25in}x = 2\,\,\,({\mbox{multiplicity 2}})}}\]