When simplifying fractions ?Do we make use of prime factorization ?

Does this below list cover all the methods of Factoring ?

Factoring polynomials

’To factor’ means to break up into multiples.

Factors of natural numbers

You will remember what you learnt about factors in Class VI. Let us take a natural number,

say 30, and write it as a product of other natural numbers, say

30 = 2 × 15

= 3 × 10 = 5 × 6

Thus, 1, 2, 3, 5, 6, 10, 15 and 30 are the factors of 30.

Of these, 2, 3 and 5 are the prime factors of 30 (Why?)

A number written as a product of prime factors is said to

be in the prime factor form; for example, 30 written as

2 × 3 × 5 is in the prime factor form.

The prime factor form of 70 is 2 × 5 × 7.

The prime factor form of 90 is 2 × 3 × 3 × 5, and so on.

Similarly, we can express algebraic expressions as products of their factors. This is

what we shall learn to do in this chapter.

Factors of algebraic expressions

We have seen in Class VII that in algebraic expressions, terms are formed as products of

factors. For example, in the algebraic expression 5xy + 3x the term 5xy has been formed

by the factors 5, x and y, i.e.,

5xy = 5 * x * y

Observe that the factors 5, x and y of 5xy cannot further

be expressed as a product of factors. We may say that 5,

x and y are ‘prime’ factors of 5xy. In algebraic expressions,

we use the word ‘irreducible’ in place of ‘prime’. We say that

5 × x × y is the irreducible form of 5xy. Note 5 × (xy) is not

an irreducible form of 5xy, since the factor xy can be further

expressed as a product of x and y, i.e., xy = x × y.

What is Factorisation?

When we factorise an algebraic expression, we write it as a product of factors. These

factors may be numbers, algebraic variables or algebraic expressions.

Expressions like 3xy, 5x2y , 2x (y + 2), 5 (y + 1) (x + 2) are already in factor form.

Their factors can be just read off from them, as we already know.

On the other hand consider expressions like 2x + 4, 3x + 3y, x2 + 5x, x2 + 5x + 6.

It is not obvious what their factors are. We need to develop systematic methods to factorise

these expressions, i.e., to find their factors.

Methods of Factoring

Method of common factors

Factorisation by regrouping terms

Factorisation using identities

Factors of the form ( x + a) ( x + b)

Factor by Splitting

Factorise 6x2 + 17x + 5 by splitting the middle term

(By splitting method) : If we can find two numbers p and q such that

p + q = 17 and pq = 6 × 5 = 30, then we can get the factors

So, let us look for the pairs of factors of 30. Some are 1 and 30, 2 and 15, 3 and 10, 5

and 6. Of these pairs, 2 and 15 will give us p + q = 17.

So, 6x2 + 17x + 5 = 6x2 + (2 + 15)x + 5

= 6x2 + 2x + 15x + 5

= 2x(3x + 1) + 5(3x + 1)

= (3x + 1) (2x + 5)

MathHands

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