Question

If alpha and beta are the zeroes of the polynomial x2 minus 4x plus 3 find the value of alpha4beta2 plus alpha2 beta4

Answer 1

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Answer 2

Answer:

The value of given expression is 90

Step-by-step explanation:

Given: $x^2-4x+3=0$

$\alpha$ and $\beta$ are the zeros of the polynomial.

Using the formula,

Sum of the root of equation , $\alpha+\beta=-\dfrac{b}{a}$

Product of root of equation, $\alpha\cdot\beta=\dfrac{c}{a}$

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

$\alpha+\beta=4$

$\alpha \beta=3$

To calculate: $\alpha^4\beta^2+\alpha^2\beta^4$

Taking $\alpha \beta$ common

$\Rightarrow \alpha^2\beta^2(\alpha^2+\beta^2)$

$\Rightarrow \alpha^2\beta^2((\alpha+\beta)^2-2\alpha \beta)$

Substitute the value of $\alpha+\beta=4$ and $\alpha \beta=3$

$\Rightarrow 3^2((4)^2-2(3))$

$\Rightarrow 9(16-6)$

$\Rightarrow 90$

$\alpha^4\beta^2+\alpha^2\beta^4=90$

Hence, The value of given expression is 90