Question

If the LCM of $f\left(x\right)={(x+1)}^5{(x+2)}^a$ and $g\left(x\right)={(x+1)}^b{(x+2)}^a$ is ${(x+1)}^a{(x+2)}^b,$ then the value of $(a+b)$ is

• A. $5$
• B. $-15$
• C. $10$
• D. $12$

Answer 1

The correct option is C $10$

Given that $f\left(x\right)=(x+1{)}^5(x+2{)}^a$ and $g\left(x\right)=(x+1{)}^b(x+2{)}^a$

Given that LCM of $f\left(x\right),g\left(x\right)$ is $(x+1{)}^a(x+2{)}^b$

Observe $(x+1)$ powers in $f\left(x\right),g\left(x\right)$ , we can see that LCM of $5,b$ must be $a$

Observe $(x+2)$ powers in $f\left(x\right),g\left(x\right)$ , We can see that LCM of $a,a$ must be $b$

LCM of $a,a$ is $a=b$

Since $a=b$ , LCM of $5,b$ is equivalent to LCM of $5,a$ , Given that it is equal to $a$

Therefore $a$ must be equal to $5$

So the value of $a+b=5+5=10$

Therefore the correct option is $C$