Question

Find all the zeros of the given function f(x) if you know that x= -4 is one of these zeros.

f(x)= -x³ + x² + 16x - 16

Answer 1

To find all zeros of the function, use synthetic division with x = -4 to get the quotient, then apply the quadratic formula to the resulting quadratic equation to find the remaining zeros.

Step-by-step explanation:

To find all the zeros of the given function, given that x = -4 is one of the zeros, we can use synthetic division or polynomial long division to divide the polynomial by (x + 4). After finding the quotient polynomial, we can use the quadratic formula to find the remaining zeros of that quadratic equation.

The quadratic formula states:

For a quadratic equation of the form ax² + bx + c = 0, the solution for x is given by x = (-b ± √(b² - 4ac)) / (2a). We apply this to the quotient to find the remaining zeros.

Answer 2

The zeros of the function f(x) are -4, 2, and 1.

Step-by-step explanation:

Given that x = -4 is one of the zeros of the function f(x) = -x³ + x² + 16x - 16, we can use polynomial division to find the quadratic equation that corresponds to the remaining zeros.

By dividing f(x) by (x + 4), we get the quadratic equation x² - 3x + 4 = 0. We can solve this equation using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Plugging in the values a = 1, b = -3, and c = 4, we get two possible values for x: x = 2 and x = 1. Therefore, the zeros of the function f(x) are -4, 2, and 1.

This formula will provide us with the two other solutions (if they are real numbers), thus finding all possible zeros of the polynomial function.