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Sagot :
Okay, here we have this:
We need to write the following polynomial in factored form as a product of linear factors:
[tex]\begin{gathered} g\mleft(t\mright)=t^3+2t^2-10t-8 \\ =\mleft(t+4\mright)\mleft(t^2-2t-2\mright) \end{gathered}[/tex]Now, let's solve the following polynomial using the general formula for equations of the second degree:
[tex]\begin{gathered} (t^2-2t-2)=0 \\ t_{1,\: 2}=\frac{-\left(-2\right)\pm\sqrt{\left(-2\right)^2-4\cdot\:1\cdot\left(-2\right)}}{2\cdot\:1} \\ t_{1,\: 2}=\frac{-\left(-2\right)\pm\:2\sqrt{3}}{2\cdot\:1} \\ t_1=\frac{-\left(-2\right)+2\sqrt{3}}{2\cdot\:1},\: t_2=\frac{-\left(-2\right)-2\sqrt{3}}{2\cdot\:1} \\ t=1+\sqrt{3},\: t=1-\sqrt{3} \end{gathered}[/tex]Finally, we obtain the following polynomial:
[tex]g(t)=(t+4)(t-1-\sqrt{3})(t-1+\sqrt{3})[/tex]
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