Question

Find the equation connecting ‘a’ and ‘b’ in order that $2{{x}^{2}}-7{{x}^{3}}+ax+b$ may be divisible by x – 3.

Answer 1

Hint: If $\left( x+\alpha \right)$ is a factor of any polynomial f(x) then $x=-\alpha $ will be a root of this polynomial i.e. f(x). Use this relation of root and factor for a polynomial to solve the problem i.e. relation between ‘a’ and ‘b’.

Complete step-by-step answer:
As we know, the factor of any polynomial will divide the polynomial completely i.e. remainder will be 0. And the factor of any polynomial gives the roots of the polynomial by equation the factor to 0. Let us suppose we have any polynomial f(x) which will give its root by the equation
f(x)…………………..(i)
Now, let $\left( x+\alpha \right)$ be the factor of f(x), then we can divide f(x) by $\left( x+\alpha \right)$ and let the quotient be g(x). So, we can write f(x) as
$f\left( x \right)=\left( x+\alpha \right)\left( g\left( x \right) \right)......................\left( ii \right)$
Now, from equation (i) and (ii) we get
$\begin{align}
  & \left( x+\alpha \right)\left( g\left( x \right) \right)=0 \\
 & x+\alpha =0\Rightarrow g\left( x \right)=0 \\
 & x=-\alpha \Rightarrow g\left( x \right)=0 \\
\end{align}$
So, we can get roots of f(x) as $x=-\alpha $ and roots of equation g(x) = 0.
Hence. If $\left( x+\alpha \right)$ is a factor of f(x), then $x=-\alpha $ will be the root of f(x). Now, coming to the question, it is given that (x -3) divides the given polynomial in the problem i.e.
$2{{x}^{2}}-7{{x}^{3}}+ax+8$
It is given that (x – 3) will be a factor of this polynomial and hence, x = 3 will be a root of the polynomial as per the initial description in the solution. Hence, x = 3 will satisfy the given polynomial. So, let
$f\left( x \right)=2{{x}^{2}}-7{{x}^{3}}+ax+b..................\left( iii \right)$
As x = 3 will be a solution equation (iii), so f(3) should be 0. Hence, put x = 3 to the f(x) in the equation (iii) so, we get
$\begin{align}
  & f\left( 3 \right)=2{{\left( 3 \right)}^{4}}-7{{\left( 3 \right)}^{3}}+a\left( 3 \right)+b=0 \\
 & f\left( 3 \right)=2\times 81-7\times 27+3a+b=0 \\
 & f\left( 3 \right)=162-189+3a+b=0 \\
 & -27+3a+b=0 \\
 & 3a+b=27 \\
\end{align}$
Hence, equation connecting ‘a’ and ‘b’ can be given as
3a + b = 27.

Note: Remainder theorem in the polynomial chapter is given as if any function f(x) is getting divided by any factor $\left( x-\alpha \right)$ then, remainder of it can be given by relation f(x). So, one may follow this theorem as well to solve the problem.
One may go wrong if he or she puts the root of the given problem as ‘-3’ by getting confused with the factor (x – 3). So, don’t confuse with the factor and root terminologies. Root of any factor can be calculated by equating the factor to 0.