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Question

Solve each polynomial equation.$x^3-4 x^2+4 x-16=0$

Solution

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Given polynomial is

$x^3-4x^2+4x-16=0$

We can solve for $x$ by grouping and finding common factors between terms:

\begin{aligned} (x^3-4x^2)+(4x-16)&=0\\ x^2(x-4)+4(x-4)&=0\\ (x^2+4)(x-4)&=0 \end{aligned}

Now, using the zero-product property, we get:

\begin{aligned} x^2+4=0\quad&\lor\quad x-4=0\\ x^2=-4\quad&\lor\quad x=4\\ x=\pm 2i\quad&\lor\quad \boxed{x=4} \end{aligned}

Since we only want the real solutions, it follows that $x=4$ is the only solution.

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