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  • x+1 is a factor of the polynomial (A) x^{3}+x^{2}-x+1\\ (B) x^{3}+x^{2}+x+1\\ (C)x^{4}+x^{3}+x^{2}+1\\ (D)x^{4}+3x^{3}+3x^{2}+x+1

x+1 is a factor of the polynomial

\\(A) x^{3}+x^{2}-x+1\\ (B) x^{3}+x^{2}+x+1\\ (C)x^{4}+x^{3}+x^{2}+1\\ (D)x^{4}+3x^{3}+3x^{2}+x+1

Answers (1)

(B) x^{3}+x^{2}+x+1

Solution:

We know that if (x+a)  is factor of the polynomial f(x), then it always satisfies f(-a)=0

Hence (x+1)  is factor of that polynomial which satisfies f(-1)=0 .
\\(A)f(x)=x^{3}+x^{2}-x+1\\ f(-1)=(-1)^{3}+(-1)^{2}-(-1)+1\\ =-1+1+1+1\\ f(-1)=2            Not satisfied

 

\\(B)f(x)=x^{3}+x^{2}+x+1\\ f(-1)=(-1)^{3}+(-1)^{2}+(-1)+1\\ =-1+1-1+1\\ f(-1)=0                   Satisfied


\\(C)f(x)=x^{4}+x^{3}+x^{2}+1\\ f(-1)=(-1)^{4}+(-1)^{3}+(-1)^{2}+1\\ =1-1+1+1\\ f(-1)=2      Not satisfied
\\(D)f(x)=x^{4}+3x^{3}+3x^{2}+x+1\\ f(-1)=(-1)^{4}+3(-1)^{3}+3(-1)^{2}+(-1)+1\\ =1-3+3-1+1\\ f(-1)=1       Not satisfied

Hence (x+1)  is a factor of x^{3}+x^{2}+x+1

Therefore option (B) is correct

 

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infoexpert24

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