First slide

Theory of equations

Question

If $x^{2} - 3 x + 2$ is a factor of $x^{4} - a x^{2} + b = 0$ then the equation whose roots are a and b is

Moderate

A

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

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

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

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

Solution

$x^{4} - a x^{2} + b = (x^{2} - 3 x + 2) p (x)$
$= (x - 1) (x - 2) p (x)$
where $p (x)$ is a polynomial of degree 2
$∴ 1 - a + b = 0$ and $16 - 4 a + b = 0$
$\Rightarrow a = 5, b = 4$
Thus, equation whose roots are a and b is
$x^{2} - 9 x + 20 = 0$

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