If $x - 1$ and $x + 1$ are both factors of the polynomial $a x^{2} + b x^{3} + 3 x^{2} + 5 x$ and $a$ and $b$ are constants,what is the value of $a$ ?

- 3

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1

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3

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5

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Solution

The correct options are
A $- 3$
B $1$
C $3$
D $5$
Given: $(x - 1)$ and $(x + 1)$ are the factors of $a x^{2} + b x^{3} + 3 x^{2} + 5 x$
$\Rightarrow b x^{3} + (a + 3) x^{2} + 5 x = (x - 1) (x + 1)$
$\Rightarrow b x^{3} + (a + 3) x^{2} + 5 x = x^{2} - 1$
$\Rightarrow b x^{3} + (a + 3) x^{2} + 5 x - x^{2} + 1 = 0$
$\Rightarrow b x^{3} + (a + 3 - 1) x^{2} + 5 x + 1 = 0$
$\Rightarrow b x^{3} + (a + 2) x^{2} + 5 x + 1 = 0$
$\Rightarrow b = 0, a + 2 = 0$
$∴ b = 0, a = - 2$

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