Question

I
tactorize the following using appropriate
identities :
9x2 +6xy + y2
4 y2 - 4y + 1
x2-y2/100​

Answer 1

What is factorisation ?

It is a method of breaking down an polynomial expression into it's multiples or factors in such a way that when they are multipled then their product give us back the original polynomial expression .

In simple terms ,

we can say that factorisation is a method of expanding or reducing the size of the polynomial expression under the mentioned condition .

Now, let's solve the question ;

1st Answer :

Given expression :

$9 {x}^{2} + 6xy + {y}^{2}$

This expression can be factorised using an identity ,

Here is the factorisation ;

$9 {x}^{2} + 6xy + {y}^{2} \\ \\ (3x {)}^{2} + 2(3x)(y) + (y {)}^{2} \\ \\ (3x + y {)}^{2}$

Identity used :

$(x + y {)}^{2} = {x}^{2} + {y}^{2} + 2xy$

2nd Answer :

Given expression :

$4 {y}^{2} - 4y + 1$

This can be factorised by splitting the middle term,

But first multiply the coefficient of the first term with the last term .

Now try to split the middle term in such a way that their product should be equal to the last term and their sum should be equal to middle term .

Hence , here is the factorisation

$4 {y}^{2} - 4y + 1 \\ 4 {y}^{2} - 2y - 2y + 1 \\ 2y(2y - 1) - 1(2y - 1) \\ (2y - 1) (2y - 1)$

Here we are not using identity but this question can be solved using identity .

3rd Answer :

Given expression :

${x}^{2} - \frac{ {y}^{2} }{100}$

Thia can be factorised using the identity ,

Here is the factorisation

${x}^{2} - \frac{ {y}^{2} }{100} \\ \\ (x {)}^{2} - ( \frac{y}{10} {)}^{2} \\ \\ (x + \frac{y}{10} )(x - \frac{y}{10})$

Identity used :

$(x + y)(x - y) = {x}^{2} - {y}^{2}$