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.