Question

Consider the following statements
(1) $x-2$ is a factor of $x^3-3x^2+4x-4$
(2) $x+1$ is a factor of $2x^3+4x+6$
(3) $x-1$ is a factor of $x^5+x^4-x^3+x^2-x+1$.
Of these statement.

• A. $1$ is correct.
• B. $1,2$ and $3$ are correct
• C. $2$ and $3$ are correct
• D. $1$ and $3$ are correct

Answer 1

The correct option is A $1$ is correct.

1) If $x-2$ is a factor of $x^3-3x^2+4x-4$, then,

$f\left(2\right)=0$

$f\left(2\right)=(2{)}^3-3(2{)}^2+8-4=0$

Hence, statement 1 is correct.

2) If $x+1$ is a factor of $2x^3+4x-6$, then,

$f(-1)=0$

$f(-1)=2(-1{)}^3+4(-1)-6=-12\ne 0$

Hence, statement 2 is incorrect.

3) If $x-1$ is a factor of $x^5+x^4-x^3+x^2-x+1$, then,

$f\left(1\right)=0$

$f\left(1\right)=1+1-1+1-1+1\ne 0$

Hence, statement 3 is incorrect.

Thus, only statement 1 is correct.