# If x^2-5x+1=0 then what is the value of x^4-2x^3-16x^2+13x+14 ?

Dec 1, 2017

$16$

#### Explanation:

From ${x}^{2} - 5 x + 1 = 0$ we have ${x}^{2} = 5 x - 1$ then

${x}^{4} - 2 {x}^{3} - 16 {x}^{2} + 13 x + 14 \equiv {\left(5 x - 1\right)}^{2} - 2 x \left(5 x - 1\right) - 16 \left(5 x - 1\right) + 13 x + 14 = 15 {x}^{2} - 75 x + 31$

Substituting again

$15 \left(5 x - 1\right) - 75 x + 31 = 16$

Dec 6, 2017

${x}^{4} - 2 {x}^{3} - 16 {x}^{2} + 13 x + 14 = 16$

#### Explanation:

Given:

${x}^{2} - 5 x + 1 = 0$

We find:

${x}^{4} - 2 {x}^{3} - 16 {x}^{2} + 13 x + 14 = \left({x}^{2} - 5 x + 1\right) \left({x}^{2} + 3 x - 2\right) + 16$

$\textcolor{w h i t e}{{x}^{4} - 2 {x}^{3} - 16 {x}^{2} + 13 x + 14} = 0 \left({x}^{2} + 3 x - 2\right) + 16$

$\textcolor{w h i t e}{{x}^{4} - 2 {x}^{3} - 16 {x}^{2} + 13 x + 14} = 16$

Alternatively, use ${x}^{2} = 5 x - 1$ to find:

${x}^{4} - 2 {x}^{3} - 16 {x}^{2} + 13 x + 14$

$= \left(5 x - 1\right) \left(5 x - 1\right) - 2 x \left(5 x - 1\right) - 16 \left(5 x - 1\right) + 13 x + 14$

$= 25 {x}^{2} - 10 x + 1 - 10 {x}^{2} + 2 x - 80 x + 16 + 13 x + 14$

$= 15 {x}^{2} - 75 x + 31$

$= 15 \left(5 x - 1\right) - 75 x + 31$

$= 75 x - 15 - 75 x + 31$

$= 16$

Dec 6, 2017

${x}^{4} - 2 {x}^{3} - 16 {x}^{2} + 13 x + 14 = 16$

#### Explanation:

Here's an approach using matrices.

Given:

${x}^{2} - 5 x + 1 = 0$

This quadratic equation is satisfied by a matrix called the companion matrix, taking the form:

$\left(\begin{matrix}0 & \textcolor{b l u e}{- 1} \\ 1 & \textcolor{b l u e}{5}\end{matrix}\right)$

Putting $x = \left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right)$

We find:

${x}^{4} - 2 {x}^{3} - 16 {x}^{2} + 13 x + 14$

$= x \left(x \left(x \left(x - 2\right) - 16\right) + 13\right) + 14$

$= \left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) - \left(\begin{matrix}2 & 0 \\ 0 & 2\end{matrix}\right)\right) - \left(\begin{matrix}16 & 0 \\ 0 & 16\end{matrix}\right)\right) + \left(\begin{matrix}13 & 0 \\ 0 & 13\end{matrix}\right)\right) + \left(\begin{matrix}14 & 0 \\ 0 & 14\end{matrix}\right)$

$= \left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\begin{matrix}- 2 & - 1 \\ 1 & 3\end{matrix}\right) - \left(\begin{matrix}16 & 0 \\ 0 & 16\end{matrix}\right)\right) + \left(\begin{matrix}13 & 0 \\ 0 & 13\end{matrix}\right)\right) + \left(\begin{matrix}14 & 0 \\ 0 & 14\end{matrix}\right)$

$= \left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\left(\begin{matrix}- 1 & - 3 \\ 3 & 14\end{matrix}\right) - \left(\begin{matrix}16 & 0 \\ 0 & 16\end{matrix}\right)\right) + \left(\begin{matrix}13 & 0 \\ 0 & 13\end{matrix}\right)\right) + \left(\begin{matrix}14 & 0 \\ 0 & 14\end{matrix}\right)$

$= \left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\begin{matrix}- 17 & - 3 \\ 3 & - 2\end{matrix}\right) + \left(\begin{matrix}13 & 0 \\ 0 & 13\end{matrix}\right)\right) + \left(\begin{matrix}14 & 0 \\ 0 & 14\end{matrix}\right)$

$= \left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\left(\begin{matrix}- 3 & 2 \\ - 2 & - 13\end{matrix}\right) + \left(\begin{matrix}13 & 0 \\ 0 & 13\end{matrix}\right)\right) + \left(\begin{matrix}14 & 0 \\ 0 & 14\end{matrix}\right)$

$= \left(\begin{matrix}0 & - 1 \\ 1 & 5\end{matrix}\right) \left(\begin{matrix}10 & 2 \\ - 2 & 0\end{matrix}\right) + \left(\begin{matrix}14 & 0 \\ 0 & 14\end{matrix}\right)$

$= \left(\begin{matrix}2 & 0 \\ 0 & 2\end{matrix}\right) + \left(\begin{matrix}14 & 0 \\ 0 & 14\end{matrix}\right)$

$= \left(\begin{matrix}16 & 0 \\ 0 & 16\end{matrix}\right)$

$= 16$