Solving for zeros in an equation, while finding values for x with synthetic division

Question

Equation in question: y=x^4-x^3-28x^2-20x+48

Questions I am not sure how to answer:

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

2.) 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?

3.) Create a table of synthetic substitution for various values of x. (I just need to know how to format/input as text, with the answers)

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

Answer 1

From Descartes' rule of signs, we know that there can be up to 2 negative (consecutive signs the same) and 2 positive roots (consecutive signs alternate); of course, there can be an even number of complex roots (up to 4) as well.

From plotting, we see that all real roots exist, at -4, -2, 1, and 6, and all roots are singletons (they would have to be, as we only have 4 roots possible).

6 | 1 -1 -28 -20 48

6 30 12 -48

1 5 2 -8 0

-4| 1 5 2 -8

-4 -4 8

1 1 -2 0

x^2 + x - 2 = (x+2)(x-1), so our other factors are -2 and 1.