Q.Find the zeros of the polynomial function $f(x)=4x^2-25$.

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. $x^2+1$).

Conjugate irrational-zero rule. For a polynomial with rational coefficients, if $a+b\sqrt p$ ($a,b\in Q$, $p$ prime) is a zero, so is its conjugate $a-b\sqrt p$.

Synthetic division is the fast way to divide by a known linear factor $(x-a)$ once one root is already given, immediately producing the reduced polynomial (usually a quadratic) whose remaining roots can then be found by the quadratic formula or further factoring.