Question

Write the polynomial in factored form as a product of linear factors:g(t)=t^3+2t^2−10t−8

Answer 1

Okay, here we have this:

We need to write the following polynomial in factored form as a product of linear factors:

`\begin{gathered} g\mleft(t\mright)=t^3+2t^2-10t-8 \\ =\mleft(t+4\mright)\mleft(t^2-2t-2\mright) \end{gathered}`

Now, let's solve the following polynomial using the general formula for equations of the second degree:

`\begin{gathered} (t^2-2t-2)=0 \\ t_(1,\: 2)=(-\left(-2\right)\pm√(\left(-2\right)^2-4\cdot\:1\cdot\left(-2\right)))/(2\cdot\:1) \\ t_(1,\: 2)=(-\left(-2\right)\pm\:2√(3))/(2\cdot\:1) \\ t_1=(-\left(-2\right)+2√(3))/(2\cdot\:1),\: t_2=(-\left(-2\right)-2√(3))/(2\cdot\:1) \\ t=1+√(3),\: t=1-√(3) \end{gathered}`

Finally, we obtain the following polynomial:

`g(t)=(t+4)(t-1-√(3))(t-1+√(3))`