The function p is a fourth-degree polynomial with x-intercepts 1.5, 4, and 10 and y-intercept -3. If p(x) is positive only on the interval (4, 10), find p(x).

Question

This is for my college pre calculus class.

Answer 1

The x intercepts reveal the zeroes p(x)=a(x-1.5)(x-4)(x-10)(x-b).

We need to find a and b. If we multiply the factors we get a(x⁴-x³(15.5+b)+x²(61+15.5b)-x(60+61b)+60b).

If the y intercept is -3 then 60ab=-3, 20ab=-1.

p(x)≥0 in [4,10] so a=-1 and -20b=-1 and b=1/20.

p(x)=-(x⁴-15.55x³+61.775x²-63.05x+3) or:

-x⁴+15.55x³-61.775x²+63.05x-3.

Answer 2

P(2)>0

The graph extends downward in quadrant 4

Zeros are -2,1,3

Y intercept-4

Sketch the graph and determine the equation in factor form