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Higher Order Equations

If x2-3x+2 is...

Question

If $x^{2} - 3 x + 2$ is a factor of $x^{4} - a x^{2} + b$ then the equation whose roots are $a, b$ is

A

x^{2} - 9 x + 20 = 0

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B

x^{2} + 9 x - 20 = 0

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C

x^{2} - 9 x - 20 = 0

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D

x^{2} + 9 x + 20 = 0

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Solution

The correct option is A $x^{2} - 9 x + 20 = 0$
$x^{2} - 3 x + 2 = (x - 1) (x - 2)$ is factor of $f (x) = x^{4} - a x^{2} + b$
So, $f (1) = 0 and f (2) = 0$
$\Rightarrow 1 - a + b = 0 and 16 - 4 a + b = 0$
$\Rightarrow a = 5, b = 4$
$∴ Required equation: x^{2} - (5 + 4) x + 5.4 = 0$
$\Rightarrow x^{2} - 9 x + 20 = 0$

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