Question

If the polynomial \[f(x)=a{{x}^{3}}+bx-c\] is divisible by the polynomial \[g(x)={{x}^{2}}+bx+c\] , then ab=
A.1
B.\[\dfrac{1}{c}\]
C.-1
D.\[\dfrac{-1}{c}\]

Answer 1

Hint: In mathematics, a polynomial is an expression consisting of variables (also called indeterminate) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is \[{{x}^{2}}~-\text{ }4x~+\text{ }7\] . An example in three variables is \[{{x}^{3}}~+\text{ }2xy{{z}^{2}}~-~yz~+\text{ }1\] .

Complete step-by-step answer:
The most important information that would be used in this question is as follows
If the polynomial is in the form in which the coefficient of the leading term that is the term that has the highest power of the variable, is 1 then, the coefficients of the subsequent term represents the sum of the root and the constant term of the polynomial represents the product of all the roots with the appropriate negative sign when it is required. Also, the coefficients of the terms that are in between the last and the second terms also signify something or the other.
As mentioned in the question, we have to find the value of ‘ab’ using the information that has been provided in the question.
Now, as f(x) is divisible by g(x), so the roots of g(x) are also the roots of f(x).
So, let the roots of f(x) be l, m and n and the roots of g(x) be l and m.
Now, as mentioned in the hint, we can see that the sum of the roots of f(x) can be written as
\[l+m+n=0\ \ \ \ \ ...(a)\]
(Because there is no term with \[{{x}^{2}}\] )
And the product of the roots of f(x) can be written as
\[l\cdot m\cdot n=\dfrac{-c}{a}\ \ \ \ \ ...(b)\]
Similarly, the sum of the roots of g(x) can be written as
\[l+m=-b\ \ \ \ \ ...(c)\]
(Because there is no term with \[{{x}^{2}}\] )
And the product of the roots of g(x) can be written as
\[l\cdot m=c\ \ \ \ \ ...(d)\]
Now, subtracting equations (a) and (c), we get
\[\begin{align}
  & -b+n=0 \\
 & n=b\ \ \ \ \ ...(e) \\
\end{align}\]
Now, using equations (b), (d) and (e), we get
\[\begin{align}
  & c\cdot b=\dfrac{-c}{a} \\
 & a\cdot b=-1 \\
\end{align}\]
Hence, the value of ab=-1.

Note: The students can make an error if they don’t know about the fact that we can find the sum and product of all the roots through the polynomial and that is mentioned in the hint.
Also, we can get the roots of a given polynomial by hit and trial method.