Answer

Verified

420.3k+ views

**Hint:**Here we are given an equation of degree four, thus having our roots. We will find the sum and product of roots in terms of coefficients of the equation to find desired results. For equation of degree four, \[{{x}^{4}}+a{{x}^{3}}+b{{x}^{2}}+cx+d+e=0\], sum of roots is given as –

$\alpha +\beta +\gamma +\delta =-\dfrac{b}{a}$ .

Product of roots is given as –

$\alpha \beta \gamma \delta =\dfrac{e}{a}$.

Also, $\alpha \beta +\beta \gamma +\gamma \delta +\alpha \gamma +\alpha \delta +\beta \delta =\dfrac{c}{a}$ and

$\alpha \beta \gamma +\alpha \gamma \delta +\alpha \beta \delta +\gamma \beta \delta =-\dfrac{d}{a}$.

We will use these formulas for finding $\sum{{{\alpha }^{2}}\beta }$.

**Complete step-by-step solution**Before applying direct formulas and jumping to answer, let us first understand the basic formulas for ${{n}^{th}}$ polynomial.

For a polynomial of degree $n$, let roots of equation are $\alpha ,{{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}}$.

Equation in general form is given by –

$f\left( x \right)={{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+{{a}_{2}}{{x}^{n-2}}+...+{{a}_{n-1}}x+{{a}_{n}}=0$

Then,

Sum of roots, $\alpha +{{\alpha }_{1}}+{{\alpha }_{2}}+...+{{\alpha }_{n}}=\dfrac{-coefficient~~of~~{{x}^{n-1}}}{coefficient~~of~~{{x}^{n}}}$

Also, ${{\alpha }_{1}}{{\alpha }_{2}}+{{\alpha }_{1}}{{\alpha }_{3}}...={{\left( -1 \right)}^{2}}\dfrac{coefficient~~of~~{{x}^{n-2}}}{coefficient~~of~~{{x}^{n}}}$

Similarly, other formulas are:-

${{\alpha }_{1}}{{\alpha }_{2}}{{\alpha }_{3}}+{{\alpha }_{2}}{{\alpha }_{3}}{{\alpha }_{4}}...={{\left( -1 \right)}^{3}}\dfrac{coefficient~~of~~{{x}^{n-3}}}{coefficient~~of~~{{x}^{n}}}$

\[{{\alpha }_{1}}{{\alpha }_{2}}{{\alpha }_{3}}{{\alpha }_{4}}...{{\alpha }_{n}}={{\left( -1 \right)}^{n}}\dfrac{constant~~term}{coefficient~~of~~{{x}^{n}}}\]

Comparing general formulas by the general equation of degree four, \[{{x}^{4}}+a{{x}^{3}}+b{{x}^{2}}+cx+d+e=0\] having roots $\alpha ,\beta ,\gamma ,\delta $ as roots:

\[\begin{align}

& \alpha +\beta +\gamma +\delta =-\dfrac{b}{a} \\

& \alpha \beta +\beta \gamma +\gamma \delta +\alpha \gamma +\alpha \delta +\beta \delta =\dfrac{c}{a} \\

& \alpha \beta \gamma +\alpha \gamma \delta +\alpha \beta \delta +\gamma \beta \delta =-\dfrac{d}{a} \\

& \alpha \beta \gamma \delta =\dfrac{e}{a} \\

\end{align}\]

We are given the equation, \[{{x}^{4}}+a{{x}^{3}}+b{{x}^{2}}+cx+d+e=0\]. Comparing with above formulas we get –

\[\begin{align}

& \alpha +\beta +\gamma +\delta =-a~~~~~~~~~~~~~~~~~~~~~~~~~~~~...\left( 1 \right) \\

& \alpha \beta +\beta \gamma +\gamma \delta +\alpha \gamma +\alpha \delta +\beta \delta =b~~~~~...\left( 2 \right) \\

& \alpha \beta \gamma +\alpha \gamma \delta +\alpha \beta \delta +\gamma \beta \delta =-c~~~~~~~~~~~~...\left( 3 \right) \\

& \alpha \beta \gamma \delta =e \\

\end{align}\]

On multiplying $\left( 1 \right)$ and $\left( 2 \right)$, we get –

\[\begin{align}

& \left( \alpha +\beta +\gamma +\delta \right)\left( \alpha \beta +\beta \gamma +\gamma \delta +\alpha \gamma +\alpha \delta +\beta \delta \right)=-ab \\

& \Rightarrow {{\alpha }^{2}}\beta +\alpha {{\beta }^{2}}+\alpha \beta \gamma +\alpha \beta \delta +\alpha \beta \gamma +{{\beta }^{2}}\gamma +\beta {{\gamma }^{2}}+\beta \gamma \delta +\alpha \gamma \delta +\beta \gamma \delta +{{\gamma }^{2}}\delta +\gamma {{\delta }^{2}}+ \\

& {{\alpha }^{2}}\gamma +\alpha \beta \gamma +\alpha {{\gamma }^{2}}+\alpha \gamma \delta +{{\alpha }^{2}}\delta +\alpha \beta \delta +\alpha \gamma \delta +\alpha {{\delta }^{2}}+\alpha \beta \delta +{{\beta }^{2}}\delta +\gamma \beta \delta +\beta {{\delta }^{2}}=-ab \\

\end{align}\]

Rearranging the terms, we get –

\[\begin{align}

& {{\alpha }^{2}}\beta +\alpha {{\beta }^{2}}+{{\beta }^{2}}\gamma +\beta {{\gamma }^{2}}+{{\gamma }^{2}}\delta +\gamma {{\delta }^{2}}+{{\alpha }^{2}}\gamma +\alpha {{\gamma }^{2}}+{{\alpha }^{2}}\delta +\alpha {{\delta }^{2}}+{{\beta }^{2}}\delta +\beta {{\delta }^{2}}+ \\

& 3\left( \alpha \beta \gamma +\beta \gamma \delta +\alpha \beta \delta +\beta \gamma \delta \right)=-ab \\

\end{align}\]

We can write the first twelve terms by $\sum{{{\alpha }^{2}}\beta }$. Hence we get –

$\sum{{{\alpha }^{2}}\beta }+3\left( \alpha \beta \gamma +\beta \gamma \delta +\alpha \beta \delta +\beta \gamma \delta \right)=-ab$

From equation (3), we can clearly see that we can directly put the value of $-c$ in above equation. We get –

\[\begin{align}

& \sum{{{\alpha }^{2}}\beta }-3c=-ab \\

& \Rightarrow \sum{{{\alpha }^{2}}\beta }=3c-ab \\

\end{align}\]

**Hence, we have found our required answer, which is, \[\sum{{{\alpha }^{2}}\beta }=3c-ab\].**

**Note:**Students should take care of signs the most. Mistakes can be made while taking positive or negative coefficients. Easy way to remember the product of roots is that we take the positive value of the coefficient of the constant term in even polynomials and the negative value of the coefficient of the constant term in odd polynomials. As there are a lot of terms in the equation, students should do it carefully and do not skip any term. Always remember, we can check it by looking at the symmetry of terms.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Who was the first to raise the slogan Inquilab Zindabad class 8 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

One cusec is equal to how many liters class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

A resolution declaring Purna Swaraj was passed in the class 8 social science CBSE