Im not getting these answers at all, please help.
Find a formula for the fourth degree polynomial p(x) whose graph is symmetric about the y-axis (meaning it has no odd powers of x), and which has a y-intercept of 0, and global maxima at (4,512) and (−4,512).
p has maxima at 4,-4 and is 3rd degree, so
p'(x) = ax(x^2-16)
= 4ax^3 - 64ax
p(x) = a x^4 - 32ax^2
= a (x^4 - 32x^2)
p(4) = 512, so a = -2
p(x) = -2x^4 + 64x^2
To find a formula for the fourth-degree polynomial, we can start by considering the given information. The graph of the polynomial is symmetric about the y-axis, meaning it has no odd powers of x. This implies that the polynomial only contains even powers of x, such as x^2, x^4, etc.
Since the polynomial has a y-intercept of 0, we know that the constant term in the polynomial is 0. This means that there is no x^0 term in the polynomial.
Now let's find the equation for the global maxima. The global maxima occurs at two points: (4, 512) and (-4, 512). We can use this information to form two equations.
First, let's consider the point (4, 512):
512 = a(4)^4 + b(4)^2 + c
Simplifying the equation gives:
512 = 256a + 16b + c ------(1)
Second, let's consider the point (-4, 512):
512 = a(-4)^4 + b(-4)^2 + c
Simplifying the equation gives:
512 = 256a + 16b + c ------(2)
Now we have two equations with three unknowns (a, b, and c), and we need one more equation to solve for them.
Since the graph is symmetric about the y-axis, it means that the coefficients of x^4 and x^2 are the same but with opposite signs.
The equation for the fourth-degree polynomial can be written as:
p(x) = ax^4 + bx^2
Now we can apply the symmetry condition to form the third equation.
To make the coefficients of x^4 and x^2 opposite, let's introduce a negative sign in front of the term with x^2:
p(x) = ax^4 - bx^2
Plugging in x = 4 and x = -4 into the equation, we have:
512 = a(4)^4 - b(4)^2 ------(3)
512 = a(-4)^4 - b(-4)^2 ------(4)
Simplifying equations (3) and (4), we get:
(1) 512 = 256a - 16b
(2) 512 = 256a - 16b
Subtracting equation (2) from equation (1), we eliminate the variable a and solve for b:
0 = 0
Since the equation 0 = 0 is always true, it means that the value of b can be any real number.
Now that we have determined that b can be any real number, we plug it back into one of the previous equations (e.g., equation 1) to solve for a:
512 = 256a - 16b
512 = 256a - 16(0)
512 = 256a
a = 2
So, we have found the values of a and b. The polynomial becomes:
p(x) = 2x^4 - bx^2
Since the polynomial also satisfies the condition of having no odd powers of x, we have successfully found the formula for the fourth-degree polynomial p(x) that meets all the given criteria as:
p(x) = 2x^4 - bx^2