Algebra - Factorisation of polynomials Notes & Question for ICSE Class 10 2021

Syllabus for Ration & Proportion

(a) Factor Theorem.

 (b) Remainder Theorem. 

(c) Factorising a polynomial completely after obtaining one factor by factor theorem.

Quick Notes on Factorisation of polynomials

Factor Theorem

When a polynomial f (x) is divided by (x – a), the remainder is equal to f (a). If the remainder f (a) is equal to 0, then (x – a) is a factor of the polynomial f(x).

the general form is (x-a) where 'a' s zero of polynomial. It has to be in this format.

for example if x+2 i.e (x-(-2)) so here the zero is -2 

also 2x+4 , so first take 2 common 2(x+2), ignore the common part, we are left with (x+2)

so (x+2) = (x-(-2)) so the zero is -2

Example: Lets have a polynomial  x² -5x +6 divided by (x – 3)

If you find the value of f (3) = 9 – 15 + 6 = 0

Therefore, (x – 3) is a factor of the given polynomial.

If it turns out to be non zero, means its not abfactor.

Remainder Theorem

If f (x), a polynomial in x, is divided by (x – a); the remainder is equal to f (a)

For example:

If f (x) is divided by (x – 5), the remainder is f (5)

If f (x) is divided by (x + 5), the remainder is f (-5)

Factorising a polynomial completely after obtaining one factor by factor theorem.

Here generally we see a polynomial with a degree of 3 max(as in syllabus). The very first step is to find the first zero of the polynomial followed by long division.

Lets see the example to know

Question to Practice for Polynomial