# Poly function of degree n with no roots

• Dafe
In summary, the conversation discusses finding polynomial functions of degree n with n roots for even and odd values of n. The suggested function f(x) = x^n+1 works for both cases, with the roots being i and -i for even n, and -1 and two complex roots for odd n. However, it is important to note that for general n, all roots are non-real for even n and -1 is a root for odd n. The conversation also acknowledges the helpfulness of the expert's responses and suggests setting up a PayPal account for their services.
Dafe

## Homework Statement

(a) If n is even find a polynomial function of degree n with n roots.
(b) If n is odd find one with only one root.

N/A

## The Attempt at a Solution

If by no roots, they mean no real roots then I guess:
$$f(x) = x^n+1$$ would work for both even and odd n's.
The roots would be i and -i if n is even, and -1, and two complex roots if it's odd..

Suggestions are very welcome :)

Yes, that looks to me to be a perfectly good answer. However, "The roots would be i and -i if n is even, and -1, and two complex roots if it's odd" is true only for n= 2 and 3. For general n, if n is even, then all n roots are non-real, i and -i being two of the. If n is odd, -1 is a root and the other n-1 roots are non-real.

Ah, I should have stated that, thank you.
You are usually the one that answers all questions I post around here. It's incredible that you do that for free..You should set up a paypal account :)
Thanks again.

## 1. What is a polynomial function of degree n with no roots?

A polynomial function of degree n with no roots is a function in the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where n is a positive integer and all the coefficients (an, an-1, ..., a0) are real numbers. This type of polynomial function does not have any real number solutions or "roots" when solved for f(x) = 0.

## 2. How can you determine if a polynomial function has no roots?

A polynomial function has no roots if all of its coefficients are nonzero and it does not have any factors that can be factored out. In other words, the function cannot be simplified any further and does not have any real number solutions when solved for f(x) = 0.

## 3. Can a polynomial function of degree n with no roots have complex roots?

Yes, a polynomial function of degree n with no roots can have complex roots. This means that the function will have solutions when solved for f(x) = 0, but these solutions will be complex numbers rather than real numbers.

## 4. Are there any real-life applications of polynomial functions of degree n with no roots?

Yes, polynomial functions of degree n with no roots can be used to model various real-life situations such as population growth, financial investments, and motion of objects. These functions can help predict future values or behaviors based on past data.

## 5. How can you graph a polynomial function of degree n with no roots?

To graph a polynomial function of degree n with no roots, you can use a graphing calculator or plot points by substituting different x-values into the function. The resulting graph will be a curve that does not intersect the x-axis (since there are no real number solutions), but it may still have complex roots that can be found using the quadratic formula.

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