If $$\alpha ,\beta ,\gamma $$ are the zeros of the polynomial $$f(x)=a{{x}^{3}}+b{{x}^{2}}+cx+d$$ , then $${{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}=$$ $$\dfrac{{{b}^{2}}-ac}{{{a}^{2}}}$$ $$\dfrac{{{b}^{2}}-2ac}{a}$$ $$\dfrac{{{b}^{2}}+2ac}{{{b}^{2}}}$$ $$\dfrac{{{b}^{2}}-2ac}{{{a}^{2}}}$$

Question

If  $$\alpha ,\beta ,\gamma $$  are the zeros of the polynomial  $$f(x)=a{{x}^{3}}+b{{x}^{2}}+cx+d$$   , then  $${{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}=$$ $$\dfrac{{{b}^{2}}-ac}{{{a}^{2}}}$$ $$\dfrac{{{b}^{2}}-2ac}{a}$$ $$\dfrac{{{b}^{2}}+2ac}{{{b}^{2}}}$$ $$\dfrac{{{b}^{2}}-2ac}{{{a}^{2}}}$$

Vedantu Content Team · Accepted Answer

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.
Also, another important formula that is also used in this question is as follows\[{{(\alpha +\beta +\gamma )}^{2}}={{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}+2(\alpha \beta +\beta \gamma +\gamma \alpha )\]

As mentioned in the question, we have to find the value of \[{{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}\] using the information that has been provided in the question.

Now, as mentioned in the hint, we know that the last term that is the constant term of the polynomial gives the product of the roots of the polynomial, so we can write as follows
\[\alpha \beta \gamma =-\left( \dfrac{d}{a} \right)=\ \dfrac{-d}{a}\ \ \ \ ...(a)\]
The sum of all the roots of the polynomial can be written as follows
\[\alpha +\beta +\gamma =-\left( \dfrac{b}{a} \right)=\dfrac{-b}{a}\ \ \ \ \ ...(b)\]
And also, the sum of the product of roots taken two at a time can be written as follows
\[\alpha \beta +\beta \gamma +\gamma \alpha =\dfrac{c}{a}\ \ \ \ \ ...(c)\]
Now, we can simplify the expression as follows
\[\begin{align}
& ={{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}} \\
& ={{(\alpha +\beta +\gamma )}^{2}}-2(\alpha \beta +\beta \gamma +\gamma \alpha ) \\
\end{align}\]
(Using the identity that is given in the hint)
Now, we can use equations (b) and (c) in the above equation to get the result as follows
\[\begin{align}
& ={{\left( \dfrac{-b}{a} \right)}^{2}}-2\cdot \dfrac{c}{a} \\
& =\dfrac{{{b}^{2}}-2ac}{{{a}^{2}}} \\
\end{align}\]
Hence, this is the required value.
option D is the required answer.

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.

If \[\alpha ,\beta ,\gamma \] are the zeros of the polynomial \[f(x)=a{{x}^{3}}+b{{x}^{2}}+cx+d\] , then \[{{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}=\] \[\dfrac{{{b}^{2}}-ac}{{{a}^{2}}}\] \[\dfrac{{{b}^{2}}-2ac}{a}\] \[\dfrac{{{b}^{2}}+2ac}{{{b}^{2}}}\] \[\dfrac{{{b}^{2}}-2ac}{{{a}^{2}}}\]

If \[\alpha ,\beta ,\gamma \] are the zeros of the polynomial \[f(x)=a{{x}^{3}}+b{{x}^{2}}+cx+d\] , then \[{{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}=\]
\[\dfrac{{{b}^{2}}-ac}{{{a}^{2}}}\]
\[\dfrac{{{b}^{2}}-2ac}{a}\]
\[\dfrac{{{b}^{2}}+2ac}{{{b}^{2}}}\]
\[\dfrac{{{b}^{2}}-2ac}{{{a}^{2}}}\]