Question

Select the correct statment.

• A. $x+2$ is a factor of $x^2+3x+4$
• B. $x+2$ is a factor of $2x^2+4$
• C. $x+2$ is a factor of $x^4-4x^3+3x^2+5x+6$
• D. $x+2$ is a factor of $x^3+3x^2+5x+6$

Answer 1

The correct option is D $x+2$ is a factor of $x^3+3x^2+5x+6$

Factor theorm 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 $p\left(x\right)=x^3+3x^2+5x+6$
$\Rightarrow p(–2)=(-2{)}^3+3(-2{)}^2+5(-2)+6$
$=–8+12–10+6=0$
Thus, By Factor Theorem, $x+2$ is a factor of $x^3+3x^2+5x+6$.

Again, let $s\left(x\right)=x^2+3x+4$
$\Rightarrow s(-2)=4-6+4\ne 0$
Hence, not a factor

let $q\left(x\right)=x^4-4x^3+3x^2+5x+6$
$\Rightarrow q(-2)=16+32+12-10+6\ne 0$
Hence, not a factor

Simialrly let $r\left(x\right)=2x^2+4$
$\Rightarrow r(-2)=8+4\ne 0$
Hence, not a factor