Question

If $f(x) = 2{x^4} - 13{x^2} + ax + b$ is divisible by ${x^2} - 3x + 2$ the, $(a,b)$ is equal to
$\eqalign{
  & 1)\left( { - 9, - 2} \right) \cr
  & 2)(6,4) \cr
  & 3)(9,2) \cr
  & 4)(2,9) \cr} $

Answer 1

Hint: The given question has two equations. It is given that one equation is completely divisible by the other. The first equation has a few unknown variables, we need to find them. We are going to use a substitution method throughout the problem. Firstly, we are going to substitute two numbers that we get as the roots to the function given, to form two equations with just the unknown variables. Then, we can consider those two equations and solve for the unknowns.

Complete step-by-step answer:
It is given that, $f(x) = 2{x^4} - 13{x^2} + ax + b$ is divisible by ${x^2} - 3x + 2$
Let us consider, ${x^2} - 3x + 2$. By splitting the middle term, we can simplify the equation.
${x^2} - 3x + 2 = (x - 2)(x - 1)$
Let us substitute $2$ for $f(x)$
It becomes,
$\eqalign{
  & \Rightarrow f(2) = 2{(2)^4} - 13{(2)^2} + a(2) + b = 0 \cr
  & \Rightarrow 2a + b = 20.....(1) \cr} $
Now, let us substitute $1$ for $f(x)$
$\eqalign{
  & \Rightarrow f(1) = 2{(1)^4} - 13{(1)^2} + a(1) + b = 0 \cr
  & \Rightarrow a + b = 11.....(2) \cr} $
Now, let us solve the equations (1) and (2)
$\eqalign{
  & a + b = 11 \cr
  & \Rightarrow a = 11 - b \cr} $
Substituting the value of $a$in equation (1),
$\eqalign{
  & \Rightarrow 2(11 - b) + b = 20 \cr
  & \Rightarrow 22 - 2b + b = 20 \cr
  & \Rightarrow b = 2 \cr} $
Now,
$\eqalign{
  & a = 11 - b \cr
  & a = 11 - 2 \cr
  & a = 9 \cr} $
From the above equations, we find that $a = 9,b = 2$
Therefore, the final answer is $(9,2)$
Hence, option (3) gives us the correct answer.
So, the correct answer is “Option 3”.

Note: There are two equations given. The keyword in this question is that they are divisible. Therefore, they do not have a remainder when they are divided. We need to find the value of the unknowns in the first expression. Therefore, we need to find the values of the second given expression, so that it can be substituted in the first one. This is the only way to substitute. The options look similar, so be careful while choosing the right answer.