Question

If $\alpha \ and\ \beta $are the zeros of the quadratic polynomial \[f\left( t \right) = {{t}^{2}}-4t+3\], find the value of ${{\alpha }^{4}}{{\beta }^{3}}+{{\alpha }^{3}}{{\beta }^{4}}$.

Answer 1

Hint: We will be using the concept of quadratic equations to solve the problem. We will be using sum of roots and product of roots to further simplify the problem.

Complete Step-by-Step solution:
Now, we have been given $\alpha \ and\ \beta $are the zeros of the quadratic polynomial\[f\left( t \right)={{t}^{2}}-4t+3\]. We have to find the value of${{\alpha }^{4}}{{\beta }^{3}}+{{\alpha }^{3}}{{\beta }^{4}}$.
Now, to find this value we will be using the sum of roots and product of roots. We know that in a quadratic equation $a{{x}^{2}}+bx+c=0$.
$\begin{align}
  & \text{sum of roots }=\dfrac{-b}{a} \\
 & \text{product of roots }=\dfrac{c}{a} \\
\end{align}$
Therefore, in \[f\left( t \right)={{t}^{2}}-4t+3\]
$\begin{align}
  & \alpha +\beta =\text{sum of roots }=\dfrac{-\left( -4 \right)}{1}=4..........\left( 1 \right) \\
 & \alpha \beta \ \text{=}\ \text{product of roots }=\dfrac{3}{1}=3...........\left( 2 \right) \\
\end{align}$
Now, we have to find the value of ${{\alpha }^{4}}{{\beta }^{3}}+{{\alpha }^{3}}{{\beta }^{4}}$.
Now, we will take ${{\alpha }^{3}}{{\beta }^{3}}$ common in ${{\alpha }^{4}}{{\beta }^{3}}+{{\alpha }^{3}}{{\beta }^{4}}$. So, we have,
$\begin{align}
  & {{\alpha }^{4}}{{\beta }^{3}}+{{\alpha }^{3}}{{\beta }^{4}}={{\alpha }^{3}}{{\beta }^{3}}\left( \alpha +\beta \right) \\
 & ={{\left( \alpha \beta \right)}^{3}}\left( \alpha +\beta \right) \\
\end{align}$
Now, we will substitute the value of $\alpha +\beta \ and\ \alpha \beta $ from (1) and (2). So, we have,
$\begin{align}
  & {{\alpha }^{4}}{{\beta }^{3}}+{{\alpha }^{3}}{{\beta }^{4}}={{\left( 3 \right)}^{3}}\left( 4 \right) \\
 & =9\times 3\times 4 \\
 & =27\times 4 \\
 & =108 \\
\end{align}$
Therefore, the value of ${{\alpha }^{4}}{{\beta }^{3}}+{{\alpha }^{3}}{{\beta }^{4}}$is 108.

Note: To solve these types of questions one must know how to find the relation between sum of roots, product of roots and coefficient of quadratic equation.
$\begin{align}
  & \text{sum of roots }=\dfrac{-b}{a} \\
 & \text{product of roots }=\dfrac{c}{a} \\
\end{align}$