Question

Factorise the following :${{\left( 3-4y-7{{y}^{2}} \right)}^{2}}-{{\left( 4y+1 \right)}^{2}}$
A. $\left( 4-7{{y}^{2}} \right)\left( 2-8y-7{{y}^{2}} \right)$
B. $\left( 7{{y}^{2}}-4 \right)\left( 2-8y-7y \right)$
C. $\left( 4-7{{y}^{2}} \right)\left( 7{{y}^{2}}+8y-2 \right)$
D. $\left( 7{{y}^{2}}-4 \right)\left( 7{{y}^{2}}-8y-2 \right)$

Answer 1

Hint: We will be using the concepts of polynomials and also using the concept of factorization to solve the problem further. By using some algebraic identities like,
${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ solve the question.

Complete step-by-step answer:
Now, we have to factorize,
${{\left( 3-4y-7{{y}^{2}} \right)}^{2}}-{{\left( 4y+1 \right)}^{2}}$
Now, we know the algebraic identity that,
${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$
Comparing with given equation we get ${a={\left( 3-4y-7{{y}^{2}}\right)}}$ , ${b={\left( 4y+1 \right)}}$
Therefore, we have,
$\left( 3-4y-7{{y}^{2}}+4y+1 \right)\left( 3-4y-7{{y}^{2}}-4y-1 \right)$
Now, we will further simplify the both factors as,
$\left( -7{{y}^{2}}+4 \right)\left( 7{{y}^{2}}+8y+2 \right)$
We can write this as,
$\left( 4-7{{y}^{2}} \right)\left( 7{{y}^{2}}-8y+2 \right)$
Now, we will see that we can further factorize it or not and also that doing so will help in further simplifying the expression or not.
Now, we see that all the options given are the multiplication of two quadratic factors and we also have the similar answer. So, we will not factorize it further.
Therefore the correct option is (C) as its option is $\left( 4-7{{y}^{2}} \right)\left( 7{{y}^{2}}+8y-2 \right)$.

Note: To solve these types of questions it is important to remember the concepts of factorization. Also one should know basic algebraic identities like,
$\begin{align}
  & {{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab \\
 & {{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right) \\
 & {{a}^{2}}+{{b}^{2}}-2ab={{\left( a-b \right)}^{2}} \\
\end{align}$