Use complete sentences to explain how to use linear factors to determine the equation of a polynomial function.

To use linear factors to determine the equation of a polynomial function, follow these steps:

1. Identify the roots of the polynomial: The roots of the polynomial are the x-values where the function crosses the x-axis. These are the solutions to the equation f(x) = 0. Each root corresponds to a linear factor of the polynomial.

2. Write down the linear factors: Each root, r, corresponds to a linear factor of the form (x - r). So if your roots are r1, r2, r3, ..., rn, your linear factors are (x - r1), (x - r2), (x - r3), ..., (x - rn).

3. Multiply the linear factors together: The polynomial is the product of its linear factors. So, to get the equation of the polynomial, you simply multiply all the linear factors together. This will give you a polynomial of degree n (if you have n roots).

4. Determine the leading coefficient: The leading coefficient is the coefficient of the highest degree term in the polynomial. If the leading coefficient is not 1, you will need to multiply the entire polynomial by the leading coefficient.

5. Write down the polynomial: The polynomial is now fully determined by its roots and leading coefficient. Write down the polynomial in standard form, which is in decreasing powers of x.

Remember, the process works in reverse as well. If you have a polynomial, you can factor it to find its roots. This is often a key step in solving polynomial equations.