f(x)=(x+5)^2(x-3)^2 Need power function y= and the Behavior of the x intercept at 3 and -5?I got the x intercepts (-5,0)(3,0) and y of (0,225)

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embizze | High School Teacher | (Level 2) Educator Emeritus

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Given `f(x)=(x+5)^2(x-3)^2` :

(1) The x-intercepts are x=-5,x=3 using the zero product property

(2) The y-intercept is 225 (Substitute x=0)

(3) If you expand you will not get a power function; you will get a quartic polynomial:




(4) Both x=-5 and x=3 are minimums of the function. Note that the function is always nonnegative (the product of two nonnegative numbers) so 0 will be a minimum.

Using calculus:

(a) Use the product rule




(b) Differentiating the expanded product gives


Setting the derivative equal to zero we get:



So the critical points are -5,-1,3

Using the first derivative test we find minimums at x=-5,x=3 and a local maximum at x=-1 (The function is symmetric so the maximum occurs at the avereage of the minimums)

The graph:



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