Given:
A polynomial P is:
[tex]P(x)=x^4-x^2-2[/tex]Required:
(a) Find all zeros of P real and complex.
(b) Factor P completely.
Explanation:
The given polynomial is:
[tex]P(x)=x^{4}-x^{2}-2[/tex](a) To find the zeros of the substitute P(x)=0.
[tex]x^4-x^2-2=0[/tex]Solve it by using the middle-term splitting method.
[tex]\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}[/tex][tex]\begin{gathered} x^2-2=0 \\ x^2=2 \\ x=\pm2 \end{gathered}[/tex][tex]\begin{gathered} x^2+1=0 \\ x^2=-1 \\ x=\pm\sqrt{-1} \\ x=\pm i \end{gathered}[/tex]Thus the zeros of the given polynomial are
[tex]\sqrt{2},-\sqrt{2},i,\text{ -i}[/tex](b)
[tex]\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}[/tex]The factors of the polynomial are
[tex]\begin{equation*} (x-\sqrt{2})(x+\sqrt{2})(x^2+1) \end{equation*}[/tex]Final Answer:
(a)
[tex]\sqrt{2},-\sqrt{2},\text{ }\imaginaryI,\text{-}\imaginaryI[/tex](b)
[tex]\begin{equation*} (x-\sqrt{2})(x+\sqrt{2})(x^2+1) \end{equation*}[/tex][tex][/tex]