1)Determine the equation of the polynomial function of degree 3, with zeros -2, -1, and 4. The graph of this function passes through (5, -84).

I have no idea how to do this one.

2) When 4x^4 - a5x^2 + px + 6 is divided by 2x+1, the remainder is 2. Determine the value of p.

1) y= (x+2)(x+1)(x-4) wouldn't that work? some might want you to multiply it out.

2) Divide the polynomial by x+1/2. See if that helps

1) To find the equation of a polynomial function with given zeros, we can start by using the fact that if "a" is a zero of a polynomial, then (x - a) is a factor of the polynomial.

In this case, the given zeros are -2, -1, and 4. So, we can write the factors of the polynomial as (x + 2), (x + 1), and (x - 4).

Next, we can multiply these factors together to find the equation of the polynomial:

(x + 2) * (x + 1) * (x - 4)

Expanding this equation will give us the polynomial function of degree 3:

(x^2 + 3x + 2) * (x - 4)

Multiplying these polynomials will give us:

x^3 - 4x^2 + 3x^2 - 12x + 2x - 8

Simplifying further, we get:

x^3 - x^2 - 10x - 8

Now, we know the equation of the polynomial function. But we still need to use the fact that the graph passes through the point (5, -84) to determine the value of the leading coefficient.

To find the leading coefficient, we substitute the x and y values of the given point into the equation and solve for the leading coefficient:

(-84) = (5)^3 - (5)^2 - 10(5) - 8

Simplifying this equation:

-84 = 125 - 25 - 50 - 8

-84 = 42 - 50

-84 = -8

This shows that the leading coefficient should be -8.

Therefore, the equation of the polynomial function is:

f(x) = -8x^3 + x^2 + 10x + 8

2) To determine the value of p, we can use the remainder theorem. According to the remainder theorem, if a polynomial f(x) is divided by (x - a), then the remainder is equal to f(a).

In this case, the given polynomial is 4x^4 - a5x^2 + px + 6, and it is divided by 2x + 1 with a remainder of 2.

Using the remainder theorem, we can set up the equation:

f(-1/2) = 2

Substituting the value of x = -1/2, we get:

4(-1/2)^4 - a5(-1/2)^2 + p(-1/2) + 6 = 2

Simplifying this equation:

1/16 - a5/4 - p/2 + 6 = 2

Rearranging and combining like terms, we get:

-a5/4 - p/2 = -1/16 - 34/16

-a5/4 - p/2 = -35/16

To eliminate the fractions, we can multiply both sides of the equation by 16:

-4a5 - 8p = -35

Now, we need more information to determine the value of p.