SOLUTION: I understand Zeros but can't make this problem turn out: Identify the number of zeros for the function. p of x equals x to the third plus two x squared subtract twelve x plus 1

Algebra ->  Rational-functions -> SOLUTION: I understand Zeros but can't make this problem turn out: Identify the number of zeros for the function. p of x equals x to the third plus two x squared subtract twelve x plus 1      Log On


   

Question 274622: I understand Zeros but can't make this problem turn out:
Identify the number of zeros for the function.
p of x equals x to the third plus two x squared subtract twelve x plus 1.

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
p%28x%29+=+x%5E3%2B2x%5E2-12x%2B1
The short answer: For a polynomial function there are as many zeros as the degree of the polynomial. In this case that would be 3 (since the degree is the highest exponent of the polynomial).

The longer answer: If you are talking about zeros from the set of complex numbers (which includes all the real numbers), then the answer remains 3. But if you are talking about real zeros, then the answer is 1 or 3. This is so because zeros with an imaginary part always come in pairs. So there are either 0 or 2 zeros with an imaginary part. This means there are 3 or 1 zero with no imaginary part (i.e. real).

Here's a graph of your function:
graph%28400%2C+400%2C+-5%2C+5%2C+-30%2C+30%2C+x%5E3%2B2x%5E2-12x%2B1%29
From the graph we can see that your function has 3 real roots/zeros (because it intersects the x-axis three times. Since your function does not factor, the roots/zeros must all be irrational. From the graph you can make rough guesses as to the values of these zeros. (Note: The graph may look like it goes through the origin. But it does not.)