Question

Write a function with the given characteristics.
1.) A polynomial with rational coefficients having roots 3, 3, and 3-i

2.) A rational function with vertical asymptote x=5 and horizontal asymptote y=1/2

Answer 1

@alivejeremy

Answer 2

HEY

Answer 3

Part I:
The factors of the polynomial function (x - 3), [x-(3+i)] and [x - (3 - i)]

Now product of these factors gives the polynomial function

Answer 4

u there

Answer 5

Yea

Answer 6

oh kool

Answer 7

did it help

Answer 8

Yes, thank you!

Answer 9

Yw

Answer 10

Rational functions do represent in simple words, the division of two functions, meaning:
\[f(x)=\frac{ a_1x^n+a_2x ^{n-1}+...a _{n-1}x+a _{n}}{ b_1x^n+b_2x ^{n-1}+...+b _{n-1}x+b_n }\]
And graphically defines a hyperbola, two parabolas facing each other.
The roots must be defined by the function that is written on the numerator, meaning the function above, so, the roots will be defined by the solutions:
\[a_1x^n+a_2x ^{n-1}+...a _{n-1}x+a _{n}=0\]
asymptotes, specially the vertical and horizontal are the limitations of the domain and co-domain corresponding to the function.
This means, that the vertical asymptote will be defined by the solution or solutions for the denominator's function:
\[b_1x^n+b_2x ^{n-1}+...+b _{n-1}x+b_n = 0\]

And now for the horizontal asymptote, we can do it with limits but in essence it is by dividing both functions by the highest exponent "x":
\[Hor \left[ f(x) \right]=\frac{ \frac{ a_1x^n+a_2x ^{n-1}+...a _{n-1}x+a _{n} }{ x^n } }{ \frac{ b_1x^n+b_2x ^{n-1}+...+b _{n-1}x+b_n }{ x^n } }\]
\[Hor \left[ f(x) \right] \iff y= \frac{ a_1 }{ b_1 }\]

Answer 11

Oh, so the 2 answers above are wrong? @Owlcoffee

Answer 12

its the same

Answer 13

lol