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Question

# Which of the following polynomials has (x + 1) as a factor?I. x3+x2+x+1 II. x4+x3+x2+x+1III. x4+3x3+3x−2+x+1 II. x3+x2−(2+√2)x+√2

A
only I
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B
only III
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C
II and IV
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D
I and II
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Solution

## The correct option is B only IIf (x+1) is a factor of a polynomial P(x),Then P(−1)must be equal to zero(I) Let P(x)=x3+x2+x+1ThenP(−1)=(−1)3+(−1)2+(−1)+1=−1+1−1+1=0(||) Let P(x)=x4+x3+x2+x+1Then P(−1)=(−1)4+(−1)3+(−1)2+(−1)+1=1−1+1−1+1=1≠0(|||) Let P(x)=x4+3x3+3x2+x+2Then P(−1)=(−1)4+3(−1)3+3(−1)2+(−1)+2=1−3+3−1+1=1≠0(IV) Let p(x)=x3+x2−(2+√2)x+√2ThenP(−1)=(−1)3+(−1)2−(2+√2)(−1)+√2=−1−1+2+√2+√2=2√2≠0Hence, only (|) Is the answer

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