Question

A polynomial P is given.P(x) = x4 − x2 − 2(a) Find all zeros of P, real and complex. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)x = (b) Factor P completely.P(x) =

Answer 1

Given:

A polynomial P is:

$P(x)=x^4-x^2-2$

Required:

(a) Find all zeros of P real and complex.

(b) Factor P completely.

Explanation:

The given polynomial is:

$P(x)=x^{4}-x^{2}-2$

(a) To find the zeros of the substitute P(x)=0.

$x^4-x^2-2=0$

Solve it by using the middle-term splitting method.

$\begin{gathered} x^4-2x^2+x^2-2=0 \\ x^2(x^2-2)+1(x^2-2)=0 \\ (x^2-2)(x^2+1)=0 \end{gathered}$$\begin{gathered} x^2-2=0 \\ x^2=2 \\ x=\pm2 \end{gathered}$$\begin{gathered} x^2+1=0 \\ x^2=-1 \\ x=\pm\sqrt{-1} \\ x=\pm i \end{gathered}$

Thus the zeros of the given polynomial are

$\sqrt{2},-\sqrt{2},i,\text{ -i}$

(b)

$\begin{gathered} P(x)=x^{4}-x^{2}-2 \\ P(x)=x^4-2x^2+x^2-2 \\ P(x)=x^2(x^2-2)+1(x^2-2) \\ P(x)=(x^2-2)(x^2+1) \\ P(x)=(x-\sqrt{2})(x+\sqrt{2})(x^2+1) \end{gathered}$

The factors of the polynomial are

$\begin{equation*} (x-\sqrt{2})(x+\sqrt{2})(x^2+1) \end{equation*}$

Final Answer:

(a)

$\sqrt{2},-\sqrt{2},\text{ }\imaginaryI,\text{-}\imaginaryI$

(b)

$\begin{equation*} (x-\sqrt{2})(x+\sqrt{2})(x^2+1) \end{equation*}$