q(x) = x^4 + 3x^3 - 13x^2 - 51 - 36 / x^3 + 9x^2 + 23x + 15 = (x + 3)^2(x - 4)(x + 1) / (x + 3)(x + 5)(x + 1)
What is the domain?
What is the x-intercept(s)?
What is the y-intercept?
What is the vertical Asymptote equation(s) and coordinates of hole(s)?
Domain:
The domain of the function q(x) is all real numbers except for x = -3, -5, and -1, since these values make the denominator equal to zero.
X-intercepts:
To find the x-intercepts, set q(x) equal to zero and solve for x:
(x + 3)^2(x - 4)(x + 1) = 0
From this equation, we can see that the x-intercepts are at x = -3 (with multiplicity 2), x = 4, and x = -1.
Y-intercept:
To find the y-intercept, plug in x = 0 into the function q(x):
q(0) = (0 + 3)^2(0 - 4)(0 + 1) / (0 + 3)(0 + 5)(0 + 1)
q(0) = (3)^2(-4)(1) / (3)(5)(1)
q(0) = -36 / 15
The y-intercept is at y = -36/15.
Vertical Asymptotes and Holes:
The vertical asymptotes occur where the denominator of q(x) is equal to zero, namely at x = -3, -5, and -1.
To find the coordinates of the holes, we need to check which of the factors in the numerator cancel out the factors in the denominator at those points. At x = -3, the factors (x + 3) cancel out, so there is a hole at x = -3. At x = -5 and x = -1, the factors do not cancel out, so there are vertical asymptotes at those points. The coordinates of the hole at x = -3 are (-3, q(-3)), where q(-3) is the value of q(x) at x = -3.