$(x + 1)$ is a factor of the polynomial
(a) $x^{3} - 2 x^{2} + x + 2$ (b) $x^{3} + 2 x^{2} + x - 2$
(c) $x^{3} + 2 x^{2} - x - 2$ (d) $x^{3} + 2 x^{2} - x + 2$

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Solution

If $(x + 1)$ is a factor then remainder will be zero or $f (- 1) = 0$

(a) $f (x) = x^{3} - 2 x^{2} + x + 2 f (- 1) = (- 1)^{3} - 2 (- 1)^{2} + (- 1) + 2 = - 1 - 2 - 1 + 2 = - 2 \neq 0$

(b) $f (x) = x^{3} + 2 x^{2} + x - 2 f (- 1) = (- 1)^{3} + 2 (- 1)^{2} + (- 1) - 2 = - 1 + 2 - 1 - 2 = - 2 \neq 0$

(c) $f (x) = x^{3} + 2 x^{2} - x - 2 f (- 1) = (- 1)^{3} + 2 (- 1)^{2} - (- 1) - 2 = - 1 + 2 + 1 - 2 = 0$

(d) $f (x) = x^{3} + 2 x^{2} - x + 2 f (- 1) = (- 1)^{3} + 2 (- 1)^{2} - (- 1) + 2 = - 1 + 2 + 1 + 2 = 4 \neq 0$

So the answer is (c)

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