I would like to know if there's an efficient way to solve an expression based on an equation? For example, I have the equation:
$x^3+4x-8=0$
and the expression:
$x^7+64x^2$
which i have to find the value of.
Apparently that equation can't be solved using some common method, so i have to play a bit with the expression by squaring both sides and adding some numbers and so.
My question here is: is there an efficient method to solve these questions without guessing the steps of solution luckily ?
As mentioned in the comments, long division of polynomials works: $${x^7+64x^2}=({x^3+4x-8})\cdot p(x)+r(x)$$
First factor out $\color{red}{x^4}$, so you are left with $$(x^7+64x^2)-(x^7+4x^5-8x^4)=-4x^5+8x^4+64x^2$$
Then factor out $\color{red}{-4x^2}$, so you are left with $$(-4x^5+8x^4+64x^2)-(-4x^5-16x^3+32x^2)=8x^4+16x^3+32x^2$$
Then factor out $\color{red}{8x}$, so you are left with $$(8x^4+16x^3+32x^2)-(8x^4+32x^2-64x)=16x^3+64x$$
Finally factor out $\color{red}{16}$, so you are left with $$(16x^3+64x)-(16x^3+64x-128)=128$$ So you have that $${x^7+64x^2}=({x^3+4x-8})\color{red}{\underbrace{(x^4-4x^2+8x+16)}_{\text{the factors}}}+128$$So when $x$ is such that $x^3+4x-8=0$, you have that $x^7+64x^2=128$.