Polynomial factor-Nightmare

Roots of equation are ${ x }^{ 2 }-x-1$ are $\phi =\frac { 1+\sqrt { 5 } }{ 2 } ,\tau =\frac { 1-\sqrt { 5 } }{ 2 }$ .

As ${ x }^{ 2 }-x-1$ is factor of ${ zx }^{ 20 }+{ yx }^{ 19 }+1$ .

Therefore we get $z\phi ^{ 20 }+{ y\phi }^{ 19 }+1=0,{ z\tau }^{ 20 }+{ y\tau }^{ 19 }+1=0$

Subtracting 2nd equation from 1st we get $z\phi ^{ 20 }-{ z\tau }^{ 20 }+{ y\phi }^{ 19 }-{ y\tau }^{ 19 }=0=z(\phi ^{ 20 }-{ \tau }^{ 20 })+{ y(\phi }^{ 19 }-{ \tau }^{ 19 })$

Now we have formula ${ f }_{ n }=\frac { \left( { \phi }^{ n }-{ \tau }^{ n } \right) }{ \sqrt { 5 } }$ where ${ f }_{ n }$ is $nth$ Fibonacci number.

Using above formula we have our expression as $z{ \sqrt { 5 } f }_{ 20 }+y{ \sqrt { 5 } f }_{ 19 }=0$ .,,,,,(1)

Dividing by $\sqrt { 5 }$ we get $z{ f }_{ 20 }+y{ f }_{ 19 }=0$ .

As we know $20th$ and $19th$ Fibonacci numbers are $6765$ and $4181$ .

Now substituting in (1) we get our equation as $6765z+4181y=0$

Clearly $z=\pm 4181n,y=\mp 6765n$ for $n=0,1,2,....$

Our answer occur's at $n=1$ and $z=\pm 4181$

So our answer is $4181$ .

$g(x)=x^2-x-1=0$ $\Rightarrow x=\dfrac{1\pm\sqrt{5}}{2}$ let $\varphi=\dfrac{1+\sqrt{5}}{2}, \psi=\dfrac{1-\sqrt{5}}{2}$ $f(x)=zx^{17}+yx^{16}+1$ Since $g(x)$ is factor of $f(x)$ , so $\varphi$ and $\psi$ are also the roots of $f(x)$ $f(\varphi)=z\varphi^{20}+y\varphi^{19}+1=0$ $f(\psi)=z\psi^{12}+y\psi^{19}+1=0$ The above equations are linear equations in terms of $z$ and $y$ . so we use cross-multiplication to solve them. $\Rightarrow \dfrac{z}{\varphi^{19}-\psi^{19} }= \dfrac{1}{\varphi^{20}\psi^{19} - \varphi^{19}\psi^{20}}$ $\Rightarrow z=\dfrac{\varphi^{19}-\psi^{19}}{\varphi^{19}\psi^{19}(\varphi-\psi)}$ We know that $\psi^{19}\varphi^{19}=(-1)^{19}=1$ and also that $F_{n}=\dfrac{\varphi^{n}-\psi^{n}}{\varphi-\psi}$ and $F_{19}=4181$ Substituting these values, we get $z=-4181$ $\Rightarrow\boxed{|z|=4181}$