Help! Polynomial inequality! (Precalculus)

[stoneytootz]

Background Info: Graph the functions and on the same axis using the Desmos Graphing Calculator. Find the coordinates of the -intercept(s) and intersection points by clicking on each point.

Question: Solve the following inequalities using interval notation.
For what x-values is f(x) > g(x)? Use interval notation. (3 pts)

Using synthetic division, I found that (x+1) is a factor, however, I am left with a quotient of x^3+4x^2-8x+1, which I found is not further factorable.
How would I go about answering this particular question?

 

[stoneytootz]

An additional comment, or potential theory:
Would I simply test values on the internals of (-infinity, -1)(-1, infinity)? Including the polynomial which cannot be further factored in the evaluation of each test value? If so, the answer, I believe, is:
f(x) > g(x) at interval (-1,infinity).

 

[Dr.Peterson]

Background Info: Graph the functions View attachment 22552 and View attachment 22554on the same axis using the Desmos Graphing Calculator. Find the coordinates of the View attachment 22553-intercept(s) and intersection points by clicking on each point.

Question: Solve the following inequalities using interval notation.
For what x-values is f(x) > g(x)? Use interval notation. (3 pts)

Using synthetic division, I found that (x+1) is a factor, however, I am left with a quotient of x^3+4x^2-8x+1, which I found is not further factorable.
How would I go about answering this particular question?

But the problem doesn't ask you to solve this algebraically; it asks you to find the points of intersection using Desmos. That may be a clue that you won't be able to do it otherwise!

Just do what they say to do. It is not unusual for a problem about polynomials to be best solved by technology, and sometimes even utterly impossible to solve exactly.

On the other hand, there is a trick to the problem, because if you do only what they say, you will almost certainly miss an intersection. If you think about the degree of each polynomial, you can see that this must be true; and then you can find an easy way to discover the missing intersection, by making another graph on Desmos!

 

[Steven G]

Background Info: Graph the functions View attachment 22552 and View attachment 22554on the same axis using the Desmos Graphing Calculator. Find the coordinates of the View attachment 22553-intercept(s) and intersection points by clicking on each point.

Question: Solve the following inequalities using interval notation.
For what x-values is f(x) > g(x)? Use interval notation. (3 pts)

Using synthetic division, I found that (x+1) is a factor, however, I am left with a quotient of x^3+4x^2-8x+1, which I found is not further factorable.
How would I go about answering this particular question?

You say that you found a factor of (x+1) but failed to say for which function. Please check your work as (x+1) is not a factor of either function. You also said that a cubic equation, x^3+4x^2-8x+1, is not factorable which means it has no x-intercepts! ALL cubic equations have either 1 x-intercepts or 3 x-intercepts! So it has one or three linear factors.

 

[Dr.Peterson]

You say that you found a factor of (x+1) but failed to say for which function. Please check your work as (x+1) is not a factor of either function. You also said that a cubic equation, x^3+4x^2-8x+1, is not factorable which means it has no x-intercepts! ALL cubic equations have either 1 x-intercepts or 3 x-intercepts! So it has one or three linear factors.

What the OP did was to turn the inequality into f(x) - g(x) > 0, and try to factor f(x) - g(x) = x^4 + 5x^3 - 4x^2 - 7x + 1. It factors as far as (x+1)(x^3 + 4x^2 - 8x + 1), and then doesn't factor further over the integers. There are three more zeros, but they are all irrational, and therefore difficult to find (though not in principle impossible).