Question

PLEASE HELP ME!!!!!!! I AM NOT GOOD AT MATH!
Given the function f(x)= \frac{x^2+7x+10}{x^2+9x+20} Describe where the function has a hole and how you found your answer.

Answer 1

Only hole of function $f(x) = \frac{x^{2}+7x+10 }{x^{2}+9x+20 }$ is at x=(-4)

Step-by-step explanation:

Given the function is $f(x) = \frac{x^{2}+7x+10 }{x^{2}+9x+20 }$

In order to find holes of any function, you should find when function is becoming undefined or say " infinity"

Given function is polynomial function.

It will become undefined become denominator become zero

$x^{2}+9x+20=0$

Solving for x value when denominator become zero

$x^{2}+9x+20=0\\x^{2}+5x+4x+20=0\\x(x+5)+4(x+5)=0\\(x+4)(x+5)=0$

we get possible holes at x=(-4) and x=(-5)

Check whether you can eliminate any holes

Now, Solving for x value when numerator become zero

$x^{2}+7x+10=0\\x^{2}+5x+2x+10=0\\(x+5)(x+2)=0$

x=(-5) and x=(-2)

x=(-5) is common is both numerator and denominator.

So that, we can eliminate it.

$f(x) = \frac{(x+5)(x+2)}{(x+5)(x+4)}$

$f(x) = \frac{(x+2)}{(x+4)}$

Therefore, Only hole of function $f(x) = \frac{x^{2}+7x+10 }{x^{2}+9x+20 }$ is at x=(-4)