Mathematics
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) x3 - x2 - (2 + )x +
Polynomials
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Answer
(i) x3 + x2 + x + 1
⇒ x + 1 = 0
⇒ x = -1
p(x) = x3 + x2 + x + 1
p(-1) = (-1)3 + (-1)2 + (-1) + 1
= -1 + 1 -1 + 1
= 0
Remainder is zero (0), so (x + 1) is factor of this polynomial.
(ii) x4 + x3 + x2 + x + 1
⇒ x + 1 = 0
⇒ x = -1
p(x) = x4 + x3 + x2 + x + 1
p(-1) = (-1)4 + (-1)3 + (-1)2 + (-1) + 1
= 1 -1 + 1 -1 + 1
= 1
Remainder is not zero (0), so (x + 1) is not a factor of this polynomial.
(iii) x4 + 3x3 + 3x2 + x + 1
⇒ x + 1 = 0
⇒ x = -1
p(x) = x4 + 3x3 + 3x2 + x + 1
p(-1) = (-1)4 + 3 x (-1)3 + 3 x (-12) + (-1) + 1
= 1 -3 + 3 -1 + 1
= 1
Remainder is not zero (0), so (x + 1) is not a factor of this polynomial.
(iv) x3 - x2 - (2 + )x +
⇒ x + 1 = 0
⇒ x = -1
p(x) = x3 - x2 - (2 + )x +
p(-1) = (-1)3 - (-1)2 - (2 + )(-1) +
= -1 - 1 + 2 + +
= -2 + 2 + 2
= 2
Remainder is not zero (0), so (x + 1) is not a factor of this polynomial.
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