Question

If f(x) = x3 – 14x2 + 43x – 30 and f(1) = 0, then find all of the zeros of f(x)
algebraically

Answer 1

To find all the zeros of the function f(x) = x^3 - 14x^2 + 43x - 30, we perform division using the known zero (x=1), resulting in a quadratic equation. We then apply the quadratic formula to determine the remaining zeros of the polynomial.

Step-by-step explanation:

The student has provided the equation f(x) = x3 – 14x2 + 43x – 30 and has indicated that f(1) = 0. Knowing that 1 is a zero of the function, we can use synthetic division or long division to factor the cubic equation into a quadratic equation and a linear factor. Once we find the quadratic component, we can use the quadratic formula, ax2 + bx + c = 0, to find the remaining zeros of the polynomial function.

Step-by-step solution:

Perform synthetic or long division using 1 as a root to get the quadratic component.

1. Once we have the quadratic equation in the form ax2 + bx + c, we can identify the coefficients a, b, and c.
2. Use the quadratic formula x = –b ± √(b2 – 4ac) / (2a) to solve for the remaining zeros.

By following these steps, we will be able to find all zeros of the given polynomial algebraically.