Question

The factors of $ 169{l^2} - 441{m^2} $ are____
a) $ (13l - 21m),(13l - 21m) $
b) $ (13l + 21m),(13l + 21m) $
c) $ (13l - 21m),(13l + 21m) $
d) $ (13l + 21m),(13l - 21m) $

Answer 1

Hint: We will use the factorization method to simplify the given expression by the formula given below, in doing so we will obtain the factors of the given expression. After that we will match our answer with the options, then we will choose the appropriate option. We will not check each option one by one by multiplying as it will consume a lot of time.
Formula used:
By applications of binomial formula we have,
 $ (a + b) \times (a - b) = {a^2} - {b^2} $

Complete step-by-step answer:
The general formulas derived from the squares of two expressions is called the binomial formula, it is given by,
 $ {(a + b)^2} = {a^2} + 2ab + {b^2} $ and $ {(a - b)^2} = {a^2} - 2ab + {b^2} $
The coefficients of each of the terms in this expression are called the binomial coefficients. These formulas have a lot of use in the factorization process, if an expression can be found identical to RHS of the two equations, we can then simplify them using these formulas.
The given expression is $ 169{l^2} - 441{m^2} $
We have $ 169 = {13^2} $ also $ 441 = {21^2} $
So now we have,
 $ 169{l^2} - 441{m^2} = {13^2}{l^2} - {21^2}{m^2} $
 $ \Rightarrow 169{l^2} - 441{m^2} = {(13l)^2} - {(21m)^2} $
Using the formula we have mentioned earlier we get,
 $ 169{l^2} - 441{m^2} = (13l - 21m)(13l + 21m) $
Therefore, the factors of $ 169{l^2} - 441{m^2} $ are $ (13l + 21m) $ and $ (13l - 21m) $
Also we can write it as, the factors of $ 169{l^2} - 441{m^2} $ are $ (13l - 21m) $ and $ (13l + 21m) $ .
So, the correct answer is “Option C AND D”.

Note: After getting the answer we could have concluded that any one of option ‘c’ and ‘d’ is correct, but as the order doesn’t matter that is in whichever way we multiply or arrange the factors their result will give us only one result.