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Question

Question 1
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) x3x2(2+2)x+2

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Solution

To have (x + 1) as a factor, substituting x = - 1 must give p(-1) = 0.
(i) x3+x2+x+1=(1)3+(1)2+(1)+1=1+11+1=0
Therefore, x+1 is a factor x3+x2+x+1.

(ii) x4+x3+x2+x+1=(1)4+(1)3+(1)2+(1)+1=11+11+1=1
Remainder is not 0. Therefore (x + 1) is not its factor.

(iii) x4+3x3+3x2+x+1=(1)4+3(1)3+3(1)2+(1)+1=13+31+1=1. Remainder is not 0.
Therefore, (x + 1) is not a factor.
(iv) x3+x2(2+2)x+2=(1)3(1)2(2+2)(1)+2=11+2+2+2=22
Remainder not 0.
Therefore, (x + 1) is not a factor.

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