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Which of the following options correctly represents the complete factored form of the polynomial F(x)= x - x2 - 4x-6?

Which of the following options correctly represents the complete factored form of-example-1

1 Answer

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Notice that:


F(3)=3^3-3^2-4\cdot3-6=27-9-12-6=27-27=0.

Therefore 3 is a root of the given polynomial.

Now, we can use this root to factor the polynomial:


F(x)=(x-3)(x^3-x^2-4x-6)/(x-3).

Using the synthetic division algorithm we get that:


(x^3-x^2-4x-6)/(x-3)=x^2+2x+2.

The roots of the above polynomial are:


\begin{gathered} x=-1+i, \\ x=-1-i\text{.} \end{gathered}

Therefore:


F(x)=\mleft(x-3\mright)(x+1+i)(x+1-i)\text{.}

Answer:


F(x)=(x-3)(x+1+i)(x+1-i)\text{.}

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