Question

h is a polynomial of degree 3 . h has a root of multiplicity 2 at u = 2 , h ( − 4 ) = 0 , and h ( 0 ) = − 11.52 . Find an algebraic equation for h .

Answer 1

A polynomial of degree 3 with a root of multiplicity 2 at u=2, and another root at -4, can be found by combining these factors and determining the remaining coefficient using the value of the function at x=0, which is -11.52.

Step-by-step explanation:

The student is asking for us to construct an algebraic equation for a polynomial function of degree 3, denoted as h. Given that there is a root of multiplicity 2 at u = 2, and also given that h passes through the points (-4, 0) and (0, -11.52), we can find an equation that represents this polynomial. Here's how you can construct the polynomial step by step:

1. Since there is a root of multiplicity 2 at u = 2, the polynomial h can be expressed as (x - 2)² times another linear factor.
2. Because h(-4) = 0, we know -4 is also a root of h. This gives us the third factor of our cubic polynomial, (x + 4).
3. Thus, the polynomial so far can be expressed as h(x) = a(x - 2)²(x + 4), where a is a constant that needs to be determined.
4. Using the point (0, -11.52) we can substitute x = 0 into our polynomial to solve for a. This results in h(0) = a(-2)²(4), and since h(0) = -11.52, we can solve for a.
5. Calculating a gives us the final form of our polynomial, which will represent the function h.

It is important to apply the quadratic formula or other algebraic methods correctly when solving for coefficients or roots as described above.