Q.Find the real roots of $x^4=16$.

A polynomial $a_nx^n+\cdots+a_0$ has degree $n$ (when $a_n\ne0$), leading coefficient $a_n$, and constant term $a_0$. Two polynomials are equal (as functions) exactly when their degrees match and every corresponding coefficient matches -- the basis of the method of undetermined coefficients.

Division algorithm. For polynomials $f,g$ with $g\ne0$: $f(x)=q(x)g(x)+r(x)$ with $\deg r<\deg g$, uniquely. When $g(x)=x-a$, the remainder is the constant $r(x)=f(a)$ -- the Remainder Theorem. Consequently $f(a)=0\iff(x-a)$ is a factor of $f(x)$ -- the Factor Theorem.

Zeros and multiplicity. If $f(x)=(x-a)^k g(x)$ with $g(a)\ne0$, $a$ is a zero of multiplicity $k$ (multiplicity 1 = 'simple root'). A degree-$n$ polynomial has AT MOST $n$ distinct real zeros -- possibly fewer, possibly none (e.g. $x^2+1$). …