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Polynomials

Factorisation of Polynomials

You know that any polynomial of the form p(a) can also be written as

If the remainder is zero, then p(a) = g(a) x h(a). That is, the polynomial p(a) is a product of two other polynomials g(a) and h(a). For example, 3a + 6a² = 3a x (1 + 2a).

It may be possible to express a polynomial as the product of two or more polynomials, in more than one form.

Methods of factorising polynomials

There are various methods of factorising a polynomial. They are,

1. Factorisation by dividing the expression by the HCF of the terms of the given expression.

2. Factorisation by grouping the terms of the expression.

3. Factorisation using identities.

Let us study these methods one after the other in detail.

Factorisation by dividing the expression by the HCF of the terms of the given expression

HCF of a polynomial is the largest monomial, which is a factor of each term of the polynomial.

We can factorise a polynomial by finding the Highest Common Factor (HCF) of the terms of the expression and then dividing each term by its HCF. HCF and the quotient obtained are the factors of the given expression.

Example 1:

Factorise 3a³ - 6a² + 3a.

Suggested answer:

Factors of 3a are 1, a, 3, 3a

Divide 3a³,-6a² and 3a by 3a, we get

HCF is one factor and the quotient is the other factor.

Example 2:

Suggested answer:

Divide the terms by their HCF,

Example 3:

Suggested answer:

HCF of 4,6,8 is 2.

HCF of b and

is b.

Steps for factorisation

Identify the HCF of the terms of the given expression.

Divide each term of the given expression by the HCF and find the quotient.

Write the given expression as a product of HCF and quotient.

Factorisation by grouping the terms of the expression

In many situations, we come across polynomials, which may not have common factors among its terms. In such cases, we group the terms of the expression in such a way that there are common factors among the terms of the groups so formed. Let us study such examples:

Example 1:

Suggested answer:

Observe that there is no factor common to all the terms.

Therefore regroup the terms of the expression.

In this expression, the first and the second terms together have a have a common factor. Similarly the third and the fourth terms have a common factor.

Can we regroup the terms in any other form and simplify?

Yes. The same terms can be regrouped as follows:

Example 2:

Suggested answer:

We cannot group mx and 3y or my and 3x as they do not have a common factor. Therefore, re-arrange (mx - 3y - my + 3x) as

(mx - my - 3y + 3x) and then factorise :

Steps for factorisation by grouping:

Rearrange the terms if necessary.

Group the given expression in such a way that each group has its common factor.

Identify the HCF of each group.

Identify the other factor.

Write the expression as a product of the common factor and the other factor.

Example 3:

Suggested answer:

Identify the HCF of the terms of the groups.

Now, observe that, (2x - 3a) is a common factor for both the groups. Therefore (2x - 3a) is one of the factors of the given expression. The other factor is identified by dividing each group by the common factor i.e. (2x - 3a).