Question

select the correct answer from each drop - down menu. (x - 2) is a factor of (x^{4}+2x^{3}-7x^{2}-8x + 12). the other factors are , , and .

Explanation:

Step1: Perform Polynomial Division or Use Synthetic Division

Since \((x - 2)\) is a factor, we can use polynomial long division or synthetic division to divide \(x^{4}+2x^{3}-7x^{2}-8x + 12\) by \((x - 2)\).

Using synthetic division for root \(x = 2\):

\[

$$\begin{array}{r|rrrrr} 2 & 1 & 2 & -7 & -8 & 12 \\ & & 2 & 8 & 2 & -12 \\ \hline & 1 & 4 & 1 & -6 & 0 \\ \end{array}$$

\]

So the quotient polynomial is \(x^{3}+4x^{2}+x - 6\).

Step2: Factor the Cubic Polynomial

Now we need to factor \(x^{3}+4x^{2}+x - 6\). Let's try to find a root of this cubic. By trial and error, we test \(x = 1\):

\(1^{3}+4(1)^{2}+1 - 6=1 + 4 + 1 - 6 = 0\). So \((x - 1)\) is a factor.

Divide \(x^{3}+4x^{2}+x - 6\) by \((x - 1)\) using synthetic division:

\[

$$\begin{array}{r|rrrr} 1 & 1 & 4 & 1 & -6 \\ & & 1 & 5 & 6 \\ \hline & 1 & 5 & 6 & 0 \\ \end{array}$$

\]

The quotient is \(x^{2}+5x + 6\).

Step3: Factor the Quadratic Polynomial

Factor \(x^{2}+5x + 6\). We know that \(x^{2}+5x + 6=(x + 2)(x + 3)\) (since \(2\times3 = 6\) and \(2 + 3 = 5\)).

Answer:

The other factors are \((x - 1)\), \((x + 2)\), and \((x + 3)\).