Question

Factorise 4x^2+9y2+16z^2+12xy-24yz-16xz​

Answer 1

Answer :

Given:

Polynomial expression is

$4 {x}^{2} + 9 {y}^{2} + 16 {z}^{2} + 12xy - 24yz - 16xz$

Required to find:

1. Factorised expression

Solution :

We can factorise this expression very easily.

Just we have to factorise it using an Identity.

Before Solving this question I would like to give you something which is helpful in factorisation.

The Identities are ;

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

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

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

Hence, The above given Identities are very useful in factorisation.

Now let's solve the question;

$4 {x}^{2} + 9 {y}^{2} + 16 {z}^{2} + 12xy - 24yz - 16xz \\ (2x {)}^{2} + (3y {)}^{2} + (4z {)}^{2} + 2(2x)(3y) - 2(3y)(4z) - 2(2x)(4z)$

Observe this carefully this result is not matching the identity .

This means that one of the number is negative.

Therefore let's solve this completely;

$(2x {)}^{2} + (3y {)}^{2} + ( - 4z {)}^{2} + 2(2x)(3y) + 2(3y)( - 4z) + 2(2x)( - 4z)$

However, Now it got matched with the identity;

Therefore last step is

$(2x + 3y + ( - 4z) {)}^{2}$

Therefore,

The expression is factorised..