Question

Factorize the following expression: $ 7{{a}^{2}}+14a $
(a) $ 7a\left( a+2 \right) $
(b) $ 7\left( a-2 \right) $
(c) $ 14\left( 7a+1 \right) $
(d) $ 7a\left( a-2 \right) $

Answer 1

Hint: We will see the definition of factorization. We have to take the common factors of both the terms. Then we will write the given expression as a product of this common factor and the multiple which will result in the given expression if multiplied. We will repeat this process until all such common factors have been extracted from both the terms to obtain the factorization of the given expression.

Complete step by step answer:
Factorization means that we write one number or expression as a product of multiple factors. The given expression is $ 7{{a}^{2}}+14a $ . We can see that 7 divides both the terms in the following manner,
 $ \dfrac{7{{a}^{2}}}{7}={{a}^{2}} $
 $ \dfrac{14a}{7}=2a $
Therefore, 7 is a common factor of both the terms. So, we can write the given expression as follows,
 $ 7{{a}^{2}}+14a=7\left( {{a}^{2}}+2a \right) $
Next, we can see that $ a $ divides both the terms in the following manner,
 $ \dfrac{{{a}^{2}}}{a}=a $
 $ \dfrac{2a}{a}=2 $
This implies that $ a $ is also a common factor of both the terms. So, we can write the given expression in the following manner,
 $ 7\left( {{a}^{2}}+2a \right)=7a\left( a+2 \right) $

Hence, the correct option is (a).

Note:
 Factorization is an important tool in many calculations. It allows us to break the given number or expression into smaller factors. This helps us in computing derivatives or doing integrations. We have a formula for the derivative of a product of two functions. Also, we know that integration of the product of two functions is given by the formula for integration by parts. We say that a factorization is a prime factorization when all the factors of that number or expression are prime.