calculus help me

Question

Find the sum of the parameters b and c if for them, the rational function R(x)=(x³+bx+c)/((x-1)(x-3)) does not have vertical asymptotes.

Answer 1

Hi Irakli,

If R(x) does not have vertical asymptotes, it means you have to “remove” both (x-1) and (x-3) from

R(x), meaning R(x) contains both (x-1) and (x-3) as factors, that is, there is a hole at x=1, and x=3 (instead of having vertical asymptotes at x=1, x=3).

You need to expand (x-1)(x-3) to x^2-4x+3, then using long division to divide R(x) by (x^2-4x+3) to find out what values of b and c will be to ensure remainder in the Long division is 0.

Please let me know if you have further questions.

Answer 2

Normally this function would have vertical aysmptotes at x=1 and x= 3 as these value of x would make the denominator equal to zero. Therefore we just have holes in the function which would mean that we have an (x-1) and (x-3) in the numerator . If x-1 and x-3 are factors then x=1 and x=3 are solutions to this function

Lets substitute 1 for x and we get 1+b +c =0. So b+c= -1

Lets substitute 3 for x and we get 27 + 3b +c = 0 3b + c = -27

Now we have 2 equations in 2 unknowns so we can solve for b and c

b=-13 c=12

We actually did not even need to do this since all we are asked is the sum of b + c

The top equation tells us b+c=-1