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Question

# x + 1 is a factor of the polynomial (a) x3 + x2 – x + 1 (b) x3 + x2 + x + 1 (c) x4 + x3 + x2 + 1 (d) x4 + 3x3 + 3x2 + x + 1

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Solution

## The polynomial f(x) which becomes zero when x = −1 is the required polynomial whose factor is x + 1. (a) Let f(x) = x3 + x2 – x + 1 f(−1) = (−1)3 + (−1)2 − (−1) + 1 = −1 + 1 + 1 + 1 = 2 ≠ 0 Therefore, x + 1 is not a factor of the polynomial, (b) Let f(x) = x3 + x2 + x + 1 f(−1) = (−1)3 + (−1)2 + (−1) + 1 = −1 + 1 − 1 + 1 = 0 Therefore, x + 1 is a factor of the polynomial, (c) Let f(x) = x4 + x3 + x2 + 1 f(−1) = (−1)4 + (−1)3 + (−1)2 + 1 = 1 − 1 + 1 + 1 = 2 ≠ 0 Therefore, x + 1 is not a factor of the polynomial, (d) Let f(x) = x4 + 3x3 + 3x2 + x + 1 f(−1) = (−1)4 + 3(−1)3 + 3(−1)2 + (−1) + 1 = 1 − 3 + 3 − 1 + 1 = 1 ≠ 0 Therefore, x + 1 is not a factor of the polynomial, Hence, the correct option is (b).

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