Question

find the following products you have to do 3rd part​

Answer 1

$\mathfrak{\large{\underline{\underline{Answer:-}}}}$

$\boxed{\bold{(2a - 3b - 2c)(4 {a}^{2} + 9 {b}^{2} + 4 {c}^{2} + 6ab - 6bc + 4ca) = 8 {a}^{3}- 27 {b}^{3} - 8 {c}^{3} -36abc}}$

$\mathfrak{\large{\underline{\underline{Explanation:-}}}}$

$(2a - 3b - 2c)(4 {a}^{2} + 9 {b}^{2} + 4 {c}^{2} + 6ab - 6bc + 4ca)$

It can be written as

$(2a + ( - 3b) + ( - 2c))( {(2a)}^{2} + {( - 3b)}^{2} + {( - 2c)}^{2} -2a( - 3b) - ( - 3b)( - 2c) -2a( - 2c)$

We know that, (x + y + z)(x² + y² + z² - xy - yz) = x³ + y³ + z³ - 3xyz

Here x = 2a,

y = - 3b

z = - 2c

By substituting the values in the identity we have,

$= {(2a)}^{3} + {( - 3b)}^{3} + {( - 2c)}^{3} - 3(2a)( - 3b)( - 2c)$

$= 8 {a}^{3} + ( - 27 {b}^{3}) + ( - 8 {c}^{3}) - 3(12abc)$

$= 8 {a}^{3}- 27 {b}^{3} - 8 {c}^{3} -36abc$

$\boxed{\bold{(2a - 3b - 2c)(4 {a}^{2} + 9 {b}^{2} + 4 {c}^{2} + 6ab - 6bc + 4ca) = 8 {a}^{3}- 27 {b}^{3} - 8 {c}^{3} -36abc}}$

$\mathfrak{\large{\underline{\underline{Identity\:Used:-}}}}$

(x + y + z)(x² + y² + z² - xy - yz) = x³ + y³ + z³ - 3xyz

$\mathfrak{\large{\underline{\underline{Extra\:Information:-}}}}$

What is an identity?

An equation is called an identity if it is satisfied by any value that replaces its variable.

What is an equation?

An open sentence which contains is '=' is called any equation.

What is an open sentence ?

A sentence for which we cannot decide whether it is true or false.

$\mathfrak{\large{\underline{\underline{Some\:important\:Identities:-}}}}$

1] (x + y)² = x² + y² + 2xy

2] (x - y)² = x² + y² - 2xy

3] (x + y)(x - y) = x² - y²

4] (x + a)(x + b) = x² + (a + b)x + ab