12 grade student-george

Let P(x) = x4 + ax3 + bx2 + cx + d. The graph of y = P(x) is symmetric with respect to the y-axis, has a relative max. at (0,1) and has an absolute min. at (q, -3)

a) determine the values for a, b c, and d using these values, write an equation for P(x)
b) find all possible values for q.


Hi George,

Since P(x) is symmetric in the y-axis, P(x) = P(-x) for any number x. Thus

x4 + ax3 + bx2 + cx + d = x4 - ax3 + bx2 - cx + d or ax3 + cx = 0 Since this is true for all x, substitute x = 1 and x = 2 to get two equation in a and c. Solve for a and c.

The graph of y = P(x) passes through (0,1) and hence P(0) = 1. This will tell you the value of d.

Finally (q,-3) is on the curve so P(q) = -3 and using what you know about absolute minima should give you a second equation that will allow you to solve for q.

I hope this helps,
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