Question

Which of the following polynomials has (x + 1) as a factor?
I. $x^3+x^2+x+1$ II. $x^4+x^3+x^2+x+1$
III. $x^4+3x^3+3x^{-2}+x+1$ II. $x^3+x^2-(2+\sqrt{2})x+\sqrt{2}$

• A. only I
• B. only III
• C. II and IV
• D. I and II

Answer 1

Answer: (A)

only I

If $(x+1)$ is a factor of a polynomial $P\left(x\right)$,Then $P(-1)$must be equal to zero

(I) Let P(x)=$x^3+x^2+x+1$

Then

$P(-1)=(-1{)}^3+(-1{)}^2+(-1)+1=-1+1-1+1=0$

(||) Let $P\left(x\right)=x^4+x^3+x^2+x+1$
Then

$P(-1)=(-1{)}^4+(-1{)}^3+(-1{)}^2+(-1)+1=1-1+1-1+1=1\ne 0$

(|||) Let $P\left(x\right)=x^4+3x^3+3x^2+x+2$

Then

$P(-1)=(-1{)}^4+3(-1{)}^3+3(-1{)}^2+(-1)+2=1-3+3-1+1=1\ne 0$

(IV) Let $p\left(x\right)=x^3+x^2-(2+\sqrt{2})x+\sqrt{2}$
Then

$P(-1)=(-1{)}^3+(-1{)}^2-(2+\sqrt{2})(-1)+\sqrt{2}=-1-1+2+\sqrt{2}+\sqrt{2}=2\sqrt{2}\ne 0$

Hence, only (|) Is the answer