Question

For the polynomial below, 2 is a zero.f(x) = x^3– 8x² + 18x – 124 –Express f (x) as a product of linear factors, F(x)=

Answer 1

The polynomial can be factorized in linear factors:

`\begin{gathered} f(x)=x^3-8x^2+18x-12 \\ f(x)=(x-2)((x^3-8x^2+18x-12)/(x-2)) \\ f(x)=(x-2)(x^2-6x+6) \end{gathered}`

For (x^2) - 6x - 6 we use the general equation:

`\begin{gathered} x^2-6x+6 \\ x_(1,2)=\frac{-(-6)\pm\sqrt[]{(-6)^2-4\mleft(6\mright)}}{2} \\ x_(1,2)=3\pm\frac{\sqrt[]{8}}{2}=3\pm\frac{\sqrt[]{2^3}}{2}=3\pm2^{(3)/(2)-1}=3\pm\sqrt[]{2}^{} \\ x_1=3+\sqrt[]{2} \\ x_2=3-\sqrt[]{2} \end{gathered}`

Then, our factors are:

`f(x)=(x-2)\cdot(x-(3+\sqrt[]{2}))\cdot(x-(3-\sqrt[]{2}))`