Help with these questions

Question

𝐺𝑖𝑣𝑒𝑛 𝑡ℎ𝑒 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, 𝑟(𝑥) = -2x^2+4x+30/x^2-3x-10

a. Rewrite the function in factored and reduced forms.

b. Identify the domain, location of the hole, and vertical asymptote.

c. Identify the zero, y-intercept and end behavior.

Determine the end behavior of the rational function below by providing the linear equation of the oblique

asymptote.

𝑠(𝑥) = 2x^3-x^2+4x-5/3x^2+9x

Use the following function to answer the questions that follow.

𝑝(𝑥) = x^4 − 3𝑥^3& − 9𝑥^2 + 23𝑥 − 12

a. Factor 𝑝(𝑥) completely.

Using your sketch, determine the values of 𝑥 such that 𝑝(𝑥) < 0. Provide your solution in interval

notation.

Answer 1

Hi Lee C

For your first question

r(x) =-2(x+3)/(x+2)

Since you can’t divide by zero, x cannot be -2

x≠-2 you have a vertical asymptote at x= -2 and your domain includes all real numbers except -2

When x = 0 r(x)=-3 so your y-intercept is (0,-3)

When r(x)=0 x=-3 so your x -intercept is (-3,0)

On the left branch of your curve because your coefficient is -2, as x approaches -2, y gets more positive, it keeps going upward

On the right branch of your curve as x approaches -2, y keeps getting more negative, it keeps going downward

I hope this helps and you can use it to work on the rest of your problems, please see details below

For the first rational function

r(x)= -2x²+4x+30/(x²-3x-10)

Factor the numerator and denominator completely

I will start with the expression in the numerator

-2x²+4x+30

-2(x²-2x-15)

-2(x-5)(x+3)

Next the denominator

x²-3x-10

(x-5)(x+2)

Put the factored expressions back into the original function r(x)

r(x) = -2(x-5)(x+3)/(x-5)(x+2)

(x-5) is common to the numerator and denominator so that cancels out and we are left with

r(x) =-2(x+3)/(x+2)

Since you can’t divide by zero, x cannot be -2

x≠-2 you have an asymptote at -2 and your domain includes all real numbers except -2

When x = 0 r(x)=-3 so your y-intercept is (0,-3)

When r(x)=0 x=-3 so your x -intercept is (-3,0)

On the left branch of your curve because your coefficient is -2, as x approaches -2, y gets more positive, it keeps going upward

On the right branch of your curve as x approaches -2, y keeps getting more negative, it keeps going downward

I hope this helps and you can use it to work on the rest of your problems