An expression of the form ax^{n} + bx^{n-1 }+ cx^{n-2 }+ ….+kx+ l, where each variable has a constant accompanying it as its coefficient is called a polynomial of degree ‘n’ in variable x. Thus, a polynomial is an expression in which a combination of a constant and a variable is separated by an addition or a subtraction sign. Till now we have learnt about calculating zeroes of polynomials. Zeros of polynomials, when represented in the form of another linear polynomial are known as factors of polynomials. Let us learn about various ways of factoring polynomials.

Factor theorem: For a polynomial p(x) of degree greater than or equal to one,

- x-a is a factor of p(x), if p(a) = 0
- if p(a) = 0, x-a is a factor of p(x)

Where ‘a’ is a real number

Problems related to factoring polynomials:

Question 1: Check whether x+3 is a factor of x^{3} + 3x^{2} + 5x +15?

Solution: Let, q(x) = x + 3, for calculating zero of this polynomial

x + 3 = 0

=> x = -3

Now, p(x) = x^{3} + 3x^{2} + 5x +15, let us check the value of this polynomial for x = -3,

p(-3) = (-3)^{3} + 3 (-3)^{2} + 5(-3) + 15 = -27 + 27 – 15 + 15 = 0

As, p(-3) = 0, x+3 is a factor of x^{3} + 3x^{2} + 5x +15.

Question 2: Factorize x^{2} + 5x + 6.

Solution: Let us try factorizing this polynomial using splitting the middle term method

Factoring polynomials by splitting the middle term:

In this technique we need to find two terms ‘a’ and ‘b’ such that a + b =5 and ab = 6. On solving this we obtain, a = 3 and b = 2

Thus the above expression can be written as,

x^{2} + 3x + 2x + 6 = x(x + 3) + 2(x + 3) = (x + 3)(x + 2)

Thus, x+3 and x+2 are the factors of the polynomial x^{2} + 5x + 6.