Analyze the polynomial function ​f(x)=−x^5−5x^4+25x^3+125x^2. Answer parts ​(a) through ​(e). ​[Hint: First factor the​ polynomial.]

Question

b) Find the​ x- and​ y-intercepts of the graph of the function.
The​ x-intercept(s) is/are

Answer 1

the point(s) where the graph crosses the x-axis, meaning the y-coordinate is 0. To find the x-intercepts, we set y = 0 and solve for x:

0 = -x^5 - 5x^4 + 25x^3 + 125x^2
0 = -x^2(x^3 + 5x^2 - 25x - 125)

Using synthetic division or polynomial long division, we can factor the polynomial as follows:
0 = -x^2(x + 5)(x^2 - 5)

Setting each factor to zero, we find the x-intercepts are:
x = 0, x = -5, x = ±√5

Therefore, the x-intercepts are (0, 0), (-5, 0), and (±√5, 0).

The y-intercept is the point where the graph crosses the y-axis, meaning the x-coordinate is 0. To find the y-intercept, we substitute x = 0 into the function:

f(0) = -0^5 - 5(0)^4 + 25(0)^3 + 125(0)^2
f(0) = 0

Therefore, the y-intercept is (0, 0).

In conclusion, the x-intercepts are (0, 0), (-5, 0), and (±√5, 0), and the y-intercept is (0, 0).