Introduction to factorisation using identities — lesson. Mathematics State Board, Class 7.

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What are the factors?

A factor is a number that divides the given number exactly without a remainder.

Example:

$5$ is divisible by

$1$ and

$5$ .

$20$ is divisible by

$1$ ,

$2$ ,

$4$ ,

$5$ ,

$10$ , and

$20$ .

From the above example,

$1$ and

$5$ are factors of number

$5$ .

Similarly,

$1$ ,

$2$ ,

$4$ ,

$5$ ,

$10$ , and

$20$ are factors of number

$20$ .

Factors are multiplied among themselves to form the original number.

$1 \times 5 = 5$

$1 \times 20 = 2 \times 10 = 4 \times 5 = 20$

We know that we can factorise a number.

But do you think it is possible to factorise an expression?

Yes, it is possible to factorise an expression. One of the common methods is expressions using identities.

Example:

1. General factorisation

Let us factorise the expression

$ab^3c$ .

We should expand the expression to find its factors.

On expansion, the expression

$ab^3c$ becomes

$a \times b \times b \times b \times c$ .

Therefore, the factors of the expression are

$a$ ,

$b$ , and

$c$ .

2. Factorisation using identities

Let us try to factorise the expression

$9 - y^2$ .

$9 - y^2$ can also be written as

$3^2 - y^2$ .

We know that,

$a^2 - b^2$

$= (a + b)(a - b)$ .

On applying the identity, we get:

$3^2 - y^2$

$= (3 + y)(3 - y)$

Thus, the factors of

$3^2 - y^2$ are

$(3 + y)$ and

$(3 - y)$ .

A list of common identities:

$(x + a)(x + b) = x^2 + x(a + b) + ab$

$(a + b)^2 = a^2 + 2ab + b^2$

$(a - b)^2 = a^2 - 2ab + b^2$

$(a + b)(a - b) = a^2 - b^2$

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1. Introduction to factorisation using identities

Theory: