Question

Which of the following is/are the irreducible factor(s) of $3m^2+9m+6?$

• A. 3
• B. $m+2$
• C. $m+3$
• D. $m+1$

Answer 1

Answer: (A), (B), (D)

$m+1$

Given, the expression is
$3m^2+9m+6.$
Taking 3 common from the above expression, we get
$3(m^2+3m+2)...\left(i\right)$

Now comparing $m^2+3m+2$ with the identity $x^2+(a+b)x+ab.$
$\text{We note that,}$
$(a+b)=3andab=2.$
$So,$
$2+1=3and\left(2\right)\left(1\right)=2$

Hence,
$m^2+3m+2$
$=m^2+2m+m+2$
$=m(m+2)+1(m+2)$
$=(m+2)(m+1)$

Now from (i), we get
$\Rightarrow 3m^2+9m+6=3(m^2+3m+2)=3(m+2)(m+1)$
Therefore,
3, (m+2) and (m+1) are the 3 irreducible factors of the given expression.