Question

Simplify: (l-m) (m-n) + (m-n) (n-l) – (n-l) (l-m)​

Answer 1

We have:

(l - m)(m - n) + (m - n)(n - l) – (n - l)(l - m)​

We have to simply the given expression.

Solution:

∴ (l - m)(m - n) + (m - n)(n - l) – (n - l)(l - m)​

Open the bracket, we get

= {lm - ln - $m^{2}$ + mn} + {mn - ml - $n^{2}$ + nl} - {nl - nm - $l^{2}$ + lm}

= lm - ln - $m^{2}$ + mn + mn - ml - $n^{2}$ + nl - nl + nm + $l^{2}$ - lm

=  $l^{2}$ - $m^{2}$ - $n^{2}$ - lm - nl + 3mn

∴ (l - m)(m - n) + (m - n)(n - l) – (n - l)(l - m)​ =  $l^{2}$ - $m^{2}$ - $n^{2}$ - lm - nl + 3mn

Thus, the value of the given expression is  "$l^{2}$ - $m^{2}$ - $n^{2}$ - lm - nl + 3mn".

Answer 2

Given:

(l-m) (m-n) + (m-n) (n-l) – (n-l) (l-m)​

To find:

Simplify, (l-m) (m-n) + (m-n) (n-l) – (n-l) (l-m)​

Solution:

From given, we have the data as follows.

(l - m) (m - n) + (m - n) (n - l) - (n - l) (l - m)​

= lm - ln - m² + mn + mn - ml - n² + nl - nl + nm + l² - lm

= - ln - m² + mn + mn - ml - n² + nl - nl + nm + l²

= - ln - m² + mn + mn - ml - n² + nm + l²

= - ln - m² + 3mn - ml - n² + l²

= 3mn - m² - n² + l² - l (m + n)

∴ (l - m) (m - n) + (m - n) (n - l) - (n - l) (l - m)​ = 3mn - m² - n² + l² - l (m + n)