Question

Can someone please show me how to use the Rational Root Theorem and the Fundamental Theorem of Algebra to prove my function f(x)=x³+3x²-22x-24 matches my graph.

Answer 1

here is the graph

Answer 2

@phi
@jdoe0001

Answer 3

what is the rational root theorem ?

Answer 4

http://en.wikipedia.org/wiki/Rational_root_theorem

Answer 5

that says that your rational roots would be one of the factors of 24
list the factors of 24

Answer 6

then try each factor... use synthetic division (or polynomial division) to see if that factor is a zero. if it is, then you will get a quadratic that you can factor using the quadratic formula to find the other 2 roots.

Fundamental Theorem of Algebra says that a polynomial of order 3 (highest exponent is 3) has exactly 3 roots.

Answer 7

So based on the rational root theorem, we know that the only possible roots are
-1,1, ... , -24, 24

And based on the Fundamental Theorem of Algebra, we know that degree of the function is 3 because x^3 is the highest polynomial and thus there are 3 linear factors possible, hence 3 roots on the graph.

The three roots on the graph, namely x = {4, -1, -6} corresponds to all the possible roots outlined in my drawing.

So the function matches the graph in that aspect.