How do you find the remaining zeros using synthetic division?

After performing synthetic division, you find the remaining zeros by setting the resulting quotient equal to zero and solving for x. This quotient will be a polynomial with a degree one less than the original polynomial.

Here's a breakdown of the process, incorporating information from the provided reference:

1. Perform Synthetic Division: Use synthetic division to divide the polynomial by a known zero (root). This will reduce the degree of the polynomial.

2. Write the Quotient: The result of the synthetic division gives you the coefficients of the new polynomial (the quotient). For example, the video reference mentions that if you start with an x cubed polynomial and perform synthetic division, the result will be an x squared polynomial.

3. Set the Quotient to Zero: Take the quotient polynomial and set it equal to zero. This creates an equation that you can solve for the remaining zeros.

   • Example (from reference): If after synthetic division you have 1x² + 0x + 4, you would set up the equation x² + 4 = 0.

4. Solve for x: Solve the resulting equation for x. The solutions are the remaining zeros of the original polynomial.

   • In the example above (x² + 4 = 0), you would subtract 4 from both sides (x² = -4) and then take the square root of both sides (x = ±2i). This means the remaining zeros are 2i and -2i.

In summary, synthetic division simplifies the polynomial, and then solving the resulting quotient allows you to find the remaining zeros.