Analyze the polynomial function f(x)=−x^5−5x^4+25x^3+125x^2. Answer parts (a) through (e). [Hint: First factor the polynomial.]
b) Find the x- and y-intercepts of the graph of the function.
The x-intercept(s) is/are
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).