Question

Which of the following is/are true?

• A. $x+2$ is a factor of $x^3+3x^2+5x+6$
• B. $x+2$ is not a factor of $2x+4$
• C. $x+2$ is not a factor of $x^3+3x^2+5x+6$
• D. $x+2$ is a factor of $2x+4$

Answer 1

Answer: (A), (D)

$x+2$ is a factor of $2x+4$

Factor theorem states if $p\left(x\right)$ is a polynomial of degree $n\ge 1$ and a is any real number then $x–a$ is a factor of $p\left(x\right)$, if and only if $p\left(a\right)=0$

The zero of $x+2$ is $–2.$
Let $p\left(x\right)=x^3+3x^2+5x+6$ and $s\left(x\right)=2x+4$

Then, $p(–2)=(-2{)}^3+3(-2{)}^2+5(-2)+6$
$=–8+12–10+6=0$
So, by the Factor Theorem, $x+2$ is a factor of $x^3+3x^2+5x+6$.

Again, $s(–2)=2(–2)+4=0$
So, $x+2$ is a factor of $2x+4$.