First slide

Theory of equations

Question

$If x + 1 is a factor of x^{4} + (p - 3) x^{3} - (3 p - 5) x^{2} + (2 p - 9) x + 6, then which of the following is not the other factor?$

Moderate

A

$x - 1$
B

$x - 2$
C

$x + 3$
D

None of these

Solution

$Putting x = - 1, we get - 6 p + 24 = 0 or p = 4$
$x^{4} + x^{3} - 7 x^{2} - x + 6$

$x^{4} + x^{3} - 7 x^{2} - x + 6 = (x + 1) (x^{3} - 7 x + 6)$
clearly x=1 satisfies $x^{3} - 7 x + 6$

$\begin{array}{r} = (x + 1) (x - 1) (x^{2} + x - 6) \\ = (x + 1) (x - 1) (x + 3) (x - 2) \end{array}$

$The other factors are x - 1, x - 2, x + 3 .$

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