# Determine which of the following polynomials has (x + 1) a factor:(i) (ii) (iii) (iv) To check for (x + 1) as a factor of given polynomials by remainder theorem, we check for p(-1)=0.

(i) If (x + 1) is a factor of p (x) = x3 + x2 + x + 1, p (-1) must be zero

Here, p (x) = x3 + x2 + x + 1

p (-1) = (-1)3 + (-1)2 + (-1) + 1

= -1 + 1 – 1 + 1 = 0

Therefore, x + 1 is a factor of this polynomial

(ii) If (x + 1) is a factor of p (x) = x4 + x3 + x2 + x + 1, p (-1) must be zero

Here, p (x) = x4+ x3 + x2 + x + 1

p (-1) = (-1)4 + (-1)3 + (-1)2 + (-1) + 1

= 1 - 1 + 1 – 1 + 1 ≠ 0

Therefore, x + 1 is not a factor of this polynomial

(iii) If (x + 1) is a factor of p (x) = x4 + 3x3 + 3x2 + x + 1, p (-1) must be zero

Here, p (x) = x4+ 3x3 + 3x2 + x + 1

p (-1) = (-1)4 + 3(-1)3 + 3(-1)2 + (-1) + 1

= 1 - 3 + 3 – 1 + 1 ≠ 0

Therefore, x + 1 is not a factor of this polynomial

(iv) If (x + 1) is a factor of polynomial

p (x) = x3 – x2 – (2 + √2)x + √2, p(-1) must be zero

p (-1) = (-1)3 – (-1)2 – (2 + √2) (-1) + √2

= -1 – 1 + 2 + √2 + √2

= 2√2

As, p (-1) ≠ 0

Therefore, x + 1 is not a factor of this polynomial

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