Question

If zeros of the polynomial \[f(x)={{x}^{3}}-3p{{x}^{2}}+qx-r\] are in arithmetic progression, then
A.\[2{{p}^{3}}=pq-r\]
B.\[2{{p}^{3}}=pq+r\]
C.\[{{p}^{3}}=pq-r\]
D.None of these

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 out which of the mentioned options is correct.
Now, as the three roots of the polynomial are in arithmetic progression, so, we can write the three roots as follows
a-d, a, a+d
Now, as mentioned in the hint, we know that the next term to the leading term that is the second term of the polynomial gives the sum of the roots of the polynomial, so we can write as follows
\[\begin{align}
  & \left( a-d \right)+a+\left( a+d \right)=-(-3p) \\
 & 3a=3p \\
 & a=p \\
\end{align}\]
Now, by using the value of a, we can find the following as
\[\begin{align}
  & \left( a-d \right)\cdot a\cdot \left( a+d \right)=-(-r) \\
 & {{a}^{3}}-a{{d}^{2}}=r \\
 & a({{a}^{2}}-{{d}^{2}})=r\ \ \ \ \ ...(a) \\
\end{align}\]
Now, putting the value of a in equation (a), we get
\[\begin{align}
  & a({{a}^{2}}-{{d}^{2}})=r\ \\
 & {{a}^{2}}-{{d}^{2}}=\frac{r}{a} \\
 & {{a}^{2}}-{{d}^{2}}=\frac{r}{p}\ \ \ \ \ ...(b) \\
\end{align}\]
Now, we also know that the coefficient of the third term that is in this case the coefficient of \[{{x}^{2}}\] given the sum of the product of the roots of the polynomial taken two at a time. So, we can write the following
\[\begin{align}
  & \left( a-d \right)\cdot a+a\cdot \left( a+d \right)+\left( a-d \right)\cdot \left( a+d \right)=q \\
 & {{a}^{2}}-ad+{{a}^{2}}+ad+{{a}^{2}}-{{d}^{2}}=q \\
 & 3{{a}^{2}}-{{d}^{2}}=q\ \ \ \ \ ...(c) \\
\end{align}\]
Now, on subtracting equations (b) and (c), we get
\[\begin{align}
  & 2{{a}^{2}}=q-\frac{r}{p} \\
 & 2{{p}^{3}}=qp-r \\
\end{align}\]
(Using the value of a as p)
Hence, the correct option is (a).

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.