Zeros in relation to an equation

Question

a. How many possible negative real zeros, positive real zeros, and non-real zeros does this equation have? Explain how you determined this.

b. Graph the equation in your calculator or in a graphing app. What is the general shape of the graph and how does the graph’s shape relate to the function? What are the zeros of the graph?

c. Create a table of synthetic substitution for various values of x.

d. Using synthetic division, show the way to divide this equation by one of its factors. Explain each step.

Answer 1

A fourth degree polynomial has four complex zeros.

DesCartes rule of signs states that the most number of positive real zeros is equal to the number of sign changes. There are 2 sign changes. Therefore they are either 2 or 0 positive real zeros

After substituting (-x) the polynomial has two sign changes. Therefore it is either 2 of 0 negative real zeros.

By the Rational Root Theorem the possible real roots are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16,.±24, ±48.

Answer 2

Zeros are not pos or neg; they are neutral.