Question

Write a polynomial equation, P(x) , in factored form given the following requirements:

- Degree: 4
- Leading coefficient 1
- Three of the zeros are at (1,0), (-4,0) and (2,0)
- y-intercept at (0,-24)
Note: P(x) written in factored form means it is written as a product of its factors.

Answer 1

The polynomial equation P(x) in factored form with a degree of 4, leading coefficient 1, zeros at (1,0), (-4,0), and (2,0), and y-intercept at (0,-24) is P(x) = (x - 1)(x + 4)(x - 2)(x + 3).

Step-by-step explanation:

To write a polynomial equation P(x) in factored form given the requirements, we need to form factors from the provided zeros and ensure that the y-intercept condition is satisfied. Since the given zeros are at (1,0), (-4,0), and (2,0), we can write the factors as (x - 1), (x + 4), and (x - 2). However, to have a degree of 4, we need one more factor. Since the y-intercept is at (0, -24), we know P(0) = -24, and this allows us to determine the missing factor.

We start with the assumption that P(x) = (x - 1)(x + 4)(x - 2)Q(x), where Q(x) is a linear factor (ax + b). Plugging in x = 0, we get P(0) = (-1)(4)(-2)(b) = -24. Solving for b, we find that b = 3. To ensure the leading coefficient is 1, we set a = 1, making Q(x) = (x + 3).

Therefore, the required polynomial in factored form is P(x) = (x - 1)(x + 4)(x - 2)(x + 3)