    Question

# Determine which of the following polynomials has (x + 1) a factor: (i) x3 + x2 + x + 1 (ii) x4 + x3 + x2 + x + 1 (iii) x4 + 3x3 + 3x2 + x + 1 (iv) Open in App
Solution

## (i) If (x + 1) is a factor of p(x) = x3 + x2 + x + 1, then p (−1) must be zero, otherwise (x + 1) is not a factor of p(x). p(x) = x3 + x2 + x + 1 p(−1) = (−1)3 + (−1)2 + (−1) + 1 = − 1 + 1 − 1 + 1 = 0 Hence, x + 1 is a factor of this polynomial. (ii) If (x + 1) is a factor of p(x) = x4 + x3 + x2 + x + 1, then p (−1) must be zero, otherwise (x + 1) is not a factor of p(x). p(x) = x4 + x3 + x2 + x + 1 p(−1) = (−1)4 + (−1)3 + (−1)2 + (−1) + 1 = 1 − 1 + 1 −1 + 1 = 1 As p(− 1) ≠ 0, Therefore, x + 1 is not a factor of this polynomial. (iii) If (x + 1) is a factor of polynomial p(x) = x4 + 3x3 + 3x2 + x + 1, then p(−1) must be 0, otherwise (x + 1) is not a factor of this polynomial. p(−1) = (−1)4 + 3(−1)3 + 3(−1)2 + (−1) + 1 = 1 − 3 + 3 − 1 + 1 = 1 As p(−1) ≠ 0, Therefore, x + 1 is not a factor of this polynomial. (iv) If(x + 1) is a factor of polynomial p(x) = , then p(−1) must be 0, otherwise (x + 1) is not a factor of this polynomial. As p(−1) ≠ 0, Therefore, (x + 1) is not a factor of this polynomial.  Suggest Corrections  2      Similar questions
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