# Solution of an equation with polynomials

I need to find a solution of the following equation :

$$t-2 = h(t)\cdot (t^7 -128) + f(t)\cdot (t^2 -4)$$

Where $h(t)$ and $f(t)$ are polynomials. I'm thinking I need to to nullify some of the terms but I don't know how to proceed. My teacher told us we can proceed by trial and error but I can't seem to figure it out ! Can someone help me ? Thank you !

## 2 Answers

If we divide the given equation with $t-2$ we get: $$1 = h(t)(t^6+2t^5+4t^4+8t^3+16t^2+32t+64)+f(t)(t+2)\;\;\;\;\;(*)$$ If the degree of $h$ is $m$ and the degree of $f$ is $n$ then by comparing the coefficients we see that $m+6=n+1$. Let assume that $m=0$. Then $h$ is constant polynomial and since for $t=-2$ we get $$h(-2) ={1\over 64}$$ we have $h(t) ={1\over 64}$ for all $t$. So now the equation $(*)$ became: $$0= (t+2)(t^5+4t^3+16t)+64f(t)(t+2)$$ so $$f(t) =-{1\over 64}(t^5+4t^3+16t)$$

If you want to nullify a term, set it equal to zero. By doing $t^2-4 = 0$, we get $t = 2$, which seems to work.