Factorisation of Polynomials — Formula Sheet

p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0

Degree = highest power of x

f(x)=g(x)q(x)+r(x)

deg r(x)<deg g(x)

f(x)=q(x)·g(x)

x^2-5x+6=(x-2)(x-3)

f(α)=a_0α^n+a_1α^{n-1}+a_2α^{n-2}+\ldots+a_{n-1}α+a_n

f(3)=(3)^2+5(3)+4

Either r(x)=0 or deg r(x) < deg g(x).

The value of a polynomial at x=α is obtained by substituting α for x throughout the polynomial.

For a polynomial f(x)=a_0x^n+a_1x^{n-1}+...+a_{n-1}x+a_n, the value at x=α is f(α)=a_0α^n+a_1α^{n-1}+...+a_{n-1}α+a_n.

Example

for f(x)=x^2+5x+4, the value at x=3 is found by writing f(3)=(3)^2+5(3)+4.