Notes – Polynomials

A polynomial function, px , is an algebraic expression that takes the form   p(x)=anxn+an1xn1+............+a2x2+a1x+ao , an0

where the coefficients  an , an1 , an2 , ............ , a1 , a0 are real numbers, and the powers n, n1 , n2 , ........ are non- negative integers

The degree of a polynomial is the highest power of  x in the expression.


Dividing a cubic polynomial  p(x)=a3x3+a2x2+a1x+a0 by a linear polynomial  xk .

Quotient =  a3x2+b1x+bo   , Remainder = R


For any polynomial px , the remainder when divided by  xα is pα .

Q. Find the remainder when  3x4+4x22x+1 is divided by  x+2 .

Remainder =  p2=324+42222+1 = 69


xα is a factor of  px if and only if  pα=0 .

Q. Find the value of  k if  x1 is a factor of  hx=x3kx2+2x1 .

Since  x1 is a factor of  hx , so  h1=0



Given a polynomial anxn+an1xn1+............+a2x2+a1x+ao , an0

has a factor pxq if and only if  p is a factor of  an and  q is a factor of  a0 .

This result is useful in helping us guess potential factors of a given polynomial.

The polynomial: hx=x3+3x2+6x+8

can be factorised if we can find a factor (px – q) where p is a factor of 1 and q is a factor of 4.
Factors of 1 are 1 × 1 and factors of 4 are ±1 × ±4 and ±2 × ±2, so possible factors of hx  are  x±1 , x±2 and x±4 .

x=2 , gives  h2=0 . So, x+2 is one of the factors of  hx .