If p(x) is a cubic polynomial such that p(0) = 0, P(2) = -4 and P(x) is positive only when x > 4 find p(x).
To find the cubic polynomial p(x), we need to determine its coefficients. Since p(0) = 0, we know that the constant term of the polynomial is 0.
Let's assume that p(x) is a cubic polynomial of the form p(x) = ax^3 + bx^2 + cx + d.
Given that p(0) = 0, we can substitute x = 0 into the polynomial to get:
p(0) = a(0)^3 + b(0)^2 + c(0) + d
0 = d
Therefore, the value of the constant term, which is represented by d, is 0.
Now, we have p(x) = ax^3 + bx^2 + cx.
Next, we'll use the second condition, p(2) = -4, to find the value of a, b, and c. Substituting x = 2 into the equation, we have:
p(2) = a(2)^3 + b(2)^2 + c(2) = -4
8a + 4b + 2c = -4
Finally, we need to consider the condition that p(x) is positive only when x > 4. This means that the polynomial p(x) needs to have all positive coefficients for x^3, x^2, and x. Since the constant term is already 0, we just need to ensure that a, b, and c are positive.
To summarize the conditions:
1. d = 0
2. 8a + 4b + 2c = -4
3. a, b, and c > 0
Now, you can solve the system of equations to find the specific values of a, b, and c that satisfy these conditions. Once those coefficients are determined, you will have the cubic polynomial p(x).