If $α$ and $β$ are the zeros of the quadratic polynomial $f(t)\ =\ t^2\ –\ 4t\ +\ 3$, find the value of $α^4β^3\ +\ α^3β^4$.

  Given:


$α$ and $β$ are the zeros of the quadratic polynomial $f(t) =t^2-4t+3$.

To do:

Here, we have to find the value of $α^4β^3+β^4α^3$.

Solution:  

We know that, 

The standard form of a quadratic polynomial is $at^2+bt+c$, where a, b and c are constants and $a≠0$.

Comparing the given polynomial with the standard form of a quadratic polynomial, 

$a=1$, $b=-4$ and $c=3$

Sum of the roots $= α+β = \frac{-b}{a} = \frac{–(-4)}{1} = 4$.

Product of the roots $= αβ = \frac{c}{a} = \frac{3}{1}=3$.

Therefore,

$α^4β^3+β^4α^3=α^3β^3(α+β)=(αβ)^3(α+β)$

$=(3)^3(4)$

$=27\times4$

$=108$

The value of $α^4β^3+β^4α^3$ is $108$.

Updated on: 10-Oct-2022

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