Factoring Polynomials: How To Factorise

Factoring polynomials is the reverse procedure of multiplication of factors of polynomials. An expression of the form axn + bxn-1 +kcxn-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.

Zeroes of polynomials, when represented in the form of another linear polynomial are known as factors of polynomials. After factorisation of a given polynomial, if we divide the polynomial with any of its factors, the remainder will be zero. Also, in this process, we factor the polynomial by finding its greatest common factor. Now let us learn how to factorise polynomials here with examples.

Factorisation of Polynomial

The process of finding factors of a given value is called factorisation. Factors are the integers which are multiplied to produce an original number. For example, the factors of 18 are 2, 3, 6, 9 and 18, such as;

18 = 2 x 9

18 = 2 x 3 x 3

18 = 3 x 6

Similarly, in the case of polynomials, the factors are the polynomials which are multiplied to produce the original polynomial. For example, the factors of x2 + 5x + 6 is (x + 2) (x + 3). When we multiply both x +2 and x+3, then the original polynomial is generated. After factorisation, we can also find the zeros of the polynomials. In this case, zeroes are x = -2 and x = -3.

Types of Factoring polynomials

There are six different methods to factorise the polynomials. The six methods are as follows:

  • Greatest Common Factor (GCF)
  • Grouping Method
  • Sum or difference in two cubes
  • Difference in two squares method
  • General trinomials
  • Trinomial method

In this article, let us discuss the two basic methods which we are using frequently to factorise the polynomial. Those two methods are the greatest common factor method and the grouping method. Apart from these methods, we can factorise the polynomials by the use of general algebraic identities. Similarly, if the polynomial is of a quadratic expression, we can use the quadratic equation to find the roots/factor of a given expression. The formula to find the factors of the quadratic expression (ax2+bx+c) is given by:

\(x = \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)

Greatest Common Factor

We have to find out the greatest common factor, of the given polynomial to factorise it. This process is nothing but a type of reverse procedure of distributive law, such as;

p( q + r) = pq + pr

But in the case of factorisation, it is just an inverse process;

pq + pr = p(q + r)

where p is the greatest common factor.

Factoring Polynomials By Grouping

This method is also said to be factoring by pairs. Here, the given polynomial is distributed in pairs or grouped in pairs to find the zeros.  Let us take an example.

Example: Factorise x2-15x+50

Find the two numbers which when added gives -15 and when multiplied gives 50.

So, -5 and -10 are the two numbers, such that;

(-5) + (-10) = -15

(-5) x (-10) = 50

Hence, we can write the given polynomial as;

x2-5x-10x+50

x(x-5)-10(x-5)

Taking x – 5 as common factor we get;

(x-5)(x-10)

Hence, the factors are (x – 5) and (x – 10).

Factoring Using Identities

The factorisation can be done also by using algebraic identities. The most common identities used in terms of the factorisation are:

  • (a + b)2 = a2 + 2ab + b2
  • (a – b)2 = a2 – 2ab + b2
  • a2 – b2= (a + b)(a – b)

Let us see an example:

Factorise (x2 – 112)

Using the identity, we can write the above polynomial as;

(x+11) (x-11)

Factor theorem

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

  1. x-a is a factor of p(x), if p(a) = 0
  2. If p(a) = 0, then x-a is a factor of p(x)

Where ‘a’ is a real number.

Learn more here: Factor Theorem

Factoring Polynomials Examples

Question 1:

Check whether x+3 is a factor of x3 + 3x2 + 5x +15.

Solution:

Let x + 3= 0

=> x = -3

Now, p(x) = x3 + 3x2 + 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 x3 + 3x2 + 5x +15.

Question 2:

Factorize x2 + 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 numbers ‘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:

x2 + 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 x2 + 5x + 6.

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