Question

Does a polynomial's degree determine how many x-intercepts it has?

Answer 1

For instance, x^3 + 2x - 7

Does the degree (3) mean that the polynomial with have 3 x-intercepts? Or n-1 intercepts (2). Is there any correlation there?

Answer 2

I'm in a precalc course right now and the lessons aren't very good at clarifying these things.

Answer 3

Yes, there is a very nice relation between degree and x intercepts :

The number of x intercepts cannot exceed the degree

Answer 4

For example,
A quadratic cannot have more than 2 x intercepts .

Answer 5

Of course a quadratic can have 0 or 1 or 2 x intercepts. But it can never have more than 2

Answer 6

No it does not *determine* the number of actual intercepts. Any even-degree polynomial can have zero x intercepts, and any odd-degree polynomial can have one intercept. It only tells you about the max number of intercepts as ganeshie said.

Answer 7

Actually, the degree of a polynomial determines how many "solutions" a polynomial has. For example the equation x^2 + 2x + 4 has NO x intercepts but has two solutions:
-1 plus sq root (3) *1
-1 minus sq root (3) *1
It has 2 solutions but does not cross the x-axis.

Answer 8

@wolf1728 Aren't solutions of a polynomial the same thing as "zeros", which are x-intercepts?

Answer 9

x-intercepts are real, solutions can be real or imaginary.

Answer 10

Okay... what would a graph look like with imaginary solutions?

Answer 11

No. For example, use the quadratic formula to solve x^2 + 2x + 4. It will give you the answers I just posted but they don't cross the x-axis.
agent0smith just posted a good reply

Answer 12

With imaginary solutions just means the graph is completely above or below the x-axis.

Like an x^2 graph shifted up enough so it does not touch the x-axis. Graph wolf's one for example.

Answer 13

Oh! Thank you both :) very helpful. I understand now.

Answer 14

Well, I figured I'd make a graph

Answer 15

Or you can type the equation into google
https://www.google.com/search?q=x%5E2%2B2x%2B4&oq=x%5E2%2B2x%2B4&aqs=chrome..69i57.3008j0j1&sourceid=chrome&ie=UTF-8#q=x%5E2%2B2x%2B12

Answer 16

All right agent0smith - guess I'll have to put the pencils and graph paper away. LOL

Answer 17

:D

Answer 18

Haha. Thanks you guys!