# Thread: Write a polynomial, Make up a rational function, & z varies jointly as u

1. ## Write a polynomial, Make up a rational function, & z varies jointly as u

- Write a polynomial of degree 4 that has exactly 3 distinct x-intercepts and whose graph rises to the left and right.

- Make up a rational function f (x) that has vertical asymptotes at x = 2 and x = −1, a horizontal asymptote at y = 1, a y-intercept at (0, 2), and x-intercept at 4.

- z varies jointly as u and the cube of v and inversely as the square of w; z = 9 when u = 4, v = 3, and w = 2. Find w when u = 27, v = 2, and z = 8.

Thank you so kindly for the help,
Pluto89

2. Originally Posted by pluto89
- Write a polynomial of degree 4 that has exactly 3 distinct x-intercepts and whose graph rises to the left and right.
If there are three x-intercepts for a degree-4 polynomial, what must be true of the multiplicity of one of the zeroes?

If the polynomial is "up" on both ends, is the leading coefficient positive or negative?

Use this information to invent any polynomial you like that fits the requirements: three linear factors, one of which is repeated; and a positive leading coefficient.

Originally Posted by pluto89
- Make up a rational function f (x) that has vertical asymptotes at x = 2 and x = −1, a horizontal asymptote at y = 1, a y-intercept at (0, 2), and x-intercept at 4.
If there are vertical asymptotes at x = 2 and at x = -1, what two factors must be in the denominator?

If the horizontal asymptote is at y = 1, then how must the degrees of the numerator and denominator compare, and how much their leading coefficients compare?

If y = x for x = 4, what must be a factor of the numerator?

If y = 2 when x = 0, then what must be the constant term of the numerator?

Use this to create a rational function which fits the requirements.

Originally Posted by pluto89
- z varies jointly as u and the cube of v and inversely as the square of w; z = 9 when u = 4, v = 3, and w = 2. Find w when u = 27, v = 2, and z = 8.
To learn how to set up and solve variation equations, try here.