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Question

Determine which of the following polynomials have (x+1) as a factor:    [2 MARKS]
(i) x3+x2+x+1                              
(ii) x4+x3+x2+x+1


Solution

Concept: 0.5 Mark each
Application: 0.5 Mark each

(i) Let p(x)=x3+x2+x+1   
Zero of (x+1) is −1

p(−1)=(1)3+(1)21+1=1+11+1=0

Therefore, by the Factor Theorem, (x+1) is the factor of x3+x2+x+1
 

(ii) Let p(x)=x4+x3+x2+x+1 Zero of (x+1) is −1

p(1)=(1)4+(1)3+(1)21+1
       =11+11+1=10.

Therefore, by the Factor theorem, (x+1) is not the factor of x4+x3+x2+x+1

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