x⁴-3/2x³+3x²-6x-4

Solve the solution

use the factor theorem, rational zero theorem, conjugate zeros theorem, and Descartes' rule of signs.

• Determine the number of zeros for the polynomial.
• Use the rational zero theorem to find the possible rational roots/zeros. State all the possible rational roots/zeros.
• Use synthetic division to find a rational root. Repeat until the polynomial has been factored down from a 4th degree polynomial to a quadratic polynomial.
• Solve the roots/zeros for the quadratic polynomial by factoring or using the quadratic formula.
• If you find a complex root/zero, remember they come in conjugate pairs (state the theorem as proof).
• Graph the original polynomial after finding all the roots/zeros.
• The graph should show and state the real roots/zeros, and you should use the Descartes' rule of signs.
• Create a table of positive/negative values on each side of a real root/zero to determine if the graph is above/below the root/zero.
• Determine end behavior of the polynomial and find and state the y-intercept. (Remember to state it as an ordered pair and make sure it is properly displayed in your graph.)