If $a$ and $b$ are zeroes of $x^{2} - 5 x + 4$ , find the value of
$a^{2} / b + b^{2} / a$

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Solution

We have

$x^{2} - 5 x + 4 = 0$

If $a$ and $b$ are zeros,

then

sum of zeros $= a + b = - \frac{c o f f . o f x}{c o f f o f x^{2}}$

$= - \frac{- 5}{1}$

$a + b = 5$

product of roots $= a . b = \frac{c o n s t a n t}{c o f f o f x^{2}}$

$= \frac{4}{1}$

$a . b = 4$

now,

$\frac{a^{2}}{b} + \frac{b^{2}}{a}$

$\Rightarrow \frac{a^{3} + b^{3}}{a b}$

$\Rightarrow \frac{(a + b) (a^{2} + b^{2} - a b)}{a b}$

$\Rightarrow \frac{(a + b) (a^{2} + b^{2} + 2 a b - 2 a b - a b)}{a b}$

$\Rightarrow \frac{(a + b) [(a + b)^{2} - 3 a b]}{a b}$

$\Rightarrow \frac{5 [5^{2} - 3 \times 4]}{4} = \frac{5 \times [25 - 12]}{4} = \frac{65}{4}$

Hence, this is the answer.

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