Question

If the expression $\left( 125-{{x}^{3}} \right)=\left( 5-x \right)\left( {{x}^{2}}+ax+b \right)$, then the value of ‘a’ is,
A. 4
B. 2
C. – 7
D. 5

Answer 1

Hint: We will first start by using the distributive property to expand the expression in the right hand side. Then we will compare the coefficient of each variable in the left side and right side to find the value of a.

Complete step-by-step answer:
Now, we have been given the expression as $\left( 125-{{x}^{3}} \right)=\left( 5-x \right)\left( {{x}^{2}}+ax+b \right)$.
Now, we know that according to distributive law,
$\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd$
So, using this we have the expression on LHS given to us as,
\[5{{x}^{2}}+5ax+5b-{{x}^{3}}-a{{x}^{2}}-bx\]
Now, we collect the similar terms together as,
$-{{x}^{3}}+{{x}^{2}}\left( 5-a \right)+x\left( 5a-b \right)+5b$
Now, we will compare this with LHS given as $125-{{x}^{3}}$.
So, we have the coefficient of ${{x}^{2}}$ on both sides as $5-a=0$
$\Rightarrow 5=a$
Hence, we have the correct option as (D).

Note: It is important to note that to solve this question and find the value of a we have expanded the right side of the expression and compare it with left side to find the value of a in the given question the coefficient of ${{x}^{2}}$ on one side was $5-a$ and on other 0. So, we have $5-a=0$ or $a=5$.