Question

Factorise $ {\left( {a - b} \right)^2} + 8\left( {a - b} \right) + 15 $
A. $ \left( {a - b + 3} \right)\left( {a - b + 5} \right) $
B. $ \left( {a - 2b + 3} \right)\left( {a + 7b + 5} \right) $
C. $ \left( {a - b + 3} \right)\left( {a - 7b + 4} \right) $
D. $ \left( {a - 2b + 1} \right)\left( {a - b + 5} \right) $

Answer 1

Hint: First consider the term $ \left( {a - b} \right) $ as another variable x. Then the given quadratic expression becomes $ {x^2} + 8x + 15 $ . So by using the factorization method, find the factors of $ {x^2} + 8x + 15 $ . Tip for factoring, we have to divide the middle term in such a way that the product of the coefficients of the divided terms must be equal to the product of the coefficients of the first and third terms.

Complete step-by-step answer:
We are given to factorize $ {\left( {a - b} \right)^2} + 8\left( {a - b} \right) + 15 $ .
First let us consider $ \left( {a - b} \right) $ as another variable x.
By replacing $ \left( {a - b} \right) $ as x in $ {\left( {a - b} \right)^2} + 8\left( {a - b} \right) + 15 $ , we get a new expression of the form $ {x^2} + 8x + 15 $
Here the product of the coefficients of the first and last terms is $ 1 \times 15 = 15 $ . So the middle term 8x must be divided in such a way that the product of the coefficients of the divided terms must be equal to 15.
So 8x can be divided as $ 3x + 5x $ , as $ 3x + 5x = 8x $ and the product of coefficients of 3x and 5x is $ 3 \times 5 $ which is 15.
So $ {x^2} + 8x + 15 $ becomes $ {x^2} + 3x + 5x + 15 $
Taking out x common from the first two terms and 5 from the last two terms, we get
 $ \Rightarrow x\left( {x + 3} \right) + 5\left( {x + 3} \right) $
 $ \Rightarrow \left( {x + 3} \right)\left( {x + 5} \right) $
 $ \therefore {x^2} + 8x + 15 = \left( {x + 3} \right)\left( {x + 5} \right) $
On subsFactorisation of Algebraic Expressions tituting $ \left( {a - b} \right) $ in the place of x in theFactorisation of Algebraic Expressions above expression we get,
 $ \therefore {\left( {a - b} \right)^2} + 8\left( {a - b} \right) + 15 = \left( {a - b + 3} \right)\left( {a - b + 5} \right) $
 Therefore, the correct option is Option A, $ \left( {a - b + 3} \right)\left( {a - b + 5} \right) $ .
So, the correct answer is “Option A”.

Note: We can also factorize the given expression by putting $ \left( {a - b} \right) $ as it is and by following the same process. We have replaced it with x to clear out confusion and to get a clear picture on how to factorize a quadratic expression. Every quadratic expression will have exactly 2 factors. Be careful while dividing the middle term or the term with ‘x’ while factorizing.