Questions: 2024R MATH-1324-40005 11.4a Polynomial Functions Due Monday, Dec 2, 11:59pm CST CURRENT Objective Determine equation of a polynomial given key information Question Which is an equation with a degree of 4, zeros located at (4,0) and (3,0) and a y-intercept located at (0,96)? Select the correct answer below: y=(x+4)(x+3)(x+8)(x+1) y=(x+4)(x+3)(x-2)(x-1) y=(x-4)(x-3)(x+2)(x+4) y=(x-4)(x-3)(x+8)(x-1) y=(x-4)(x+3)(x+8)(x-1)

Solution Steps

To determine the correct polynomial equation, we need to consider the given zeros and the y-intercept. The zeros at (4,0) and (3,0) imply factors of (x-4) and (x-3). Since the polynomial is of degree 4, it should have two more factors. We will test each option to see which one satisfies the y-intercept at (0,96).

The polynomial is of degree 4 with zeros at \( (4,0) \) and \( (3,0) \). This gives us the factors \( (x - 4) \) and \( (x - 3) \).

The polynomial must also have a y-intercept at \( (0, 96) \). This means that when \( x = 0 \), the polynomial should evaluate to 96.

We evaluate the following polynomial options to find which ones yield a y-intercept of 96:

1. \( y = (x + 4)(x + 3)(x + 8)(x + 1) \) → \( y(0) = 96 \)
2. \( y = (x + 4)(x + 3)(x - 2)(x - 1) \) → \( y(0) = 24 \)
3. \( y = (x - 4)(x - 3)(x + 2)(x + 4) \) → \( y(0) = 96 \)
4. \( y = (x - 4)(x - 3)(x + 8)(x - 1) \) → \( y(0) = -96 \)
5. \( y = (x - 4)(x + 3)(x + 8)(x - 1) \) → \( y(0) = 96 \)

From the evaluations, the polynomials that yield a y-intercept of 96 are options 1, 3, and 5.

Final Answer