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Q: Does a polynomial with a leading term with an even exponent must have at least one real zero?

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It depends on the domain. In the complex domain, a polynomial of order n must have n solutions, although some of these may be multiple solutions. In the real domain, a polynomial of odd order must have at least one real solution, while a polynomial of even order may have no real solutions.

Not quite. The point at infinity cannot be regarded as a maximum since the value will continue to increase asymptotically. As a result no polynomial of odd degree can have a maximum. Only polynomials of an even degree whose leading coefficient is negative will have a global maximum.

There is no limit to what may be used as an exponent. Even infinity can be an exponent (though some things, like 1∞ , are not defined at all, and others are themselves infinite).

That's true if the exponent is zero. Then it doesn't even matter what the base is.

It can be any polynomial rule with integer coefficients in which there are an even number of odd coefficients.

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The degree of a polynomial refers to the largest exponent in the function for that polynomial. A degree 3 polynomial will have 3 as the largest exponent, but may also have smaller exponents. Both x^3 and x^3-x²+x-1 are degree three polynomials since the largest exponent is 4. The polynomial x^4+x^3 would not be degree three however because even though there is an exponent of 3, there is a higher exponent also present (in this case, 4).

Leading coefficient: Negative. Order: Any even integer.

It depends on the domain. In the complex domain, a polynomial of order n must have n solutions, although some of these may be multiple solutions. In the real domain, a polynomial of odd order must have at least one real solution, while a polynomial of even order may have no real solutions.

Not quite. The point at infinity cannot be regarded as a maximum since the value will continue to increase asymptotically. As a result no polynomial of odd degree can have a maximum. Only polynomials of an even degree whose leading coefficient is negative will have a global maximum.

No. Even if the answer is zero, zero is still a polynomial.

Sort of... but not entirely. Assuming the polynomial's coefficients are real, the polynomial either has as many real roots as its degree, or an even number less. Thus, a polynomial of degree 4 can have 4, 2, or 0 real roots; while a polynomial of degree 5 has either 5, 3, or 1 real roots. So, polynomial of odd degree (with real coefficients) will always have at least one real root. For a polynomial of even degree, this is not guaranteed. (In case you are interested about the reason for the rule stated above: this is related to the fact that any complex roots in such a polynomial occur in conjugate pairs; for example: if 5 + 2i is a root, then 5 - 2i is also a root.)

Any even exponent of 3 is a square number.

Assuming that you are reffering to something like this: (x - h)(x - k) = 0 x = h, x = k This is the fundamental theorem of algebra which states that is given a polynomial (multiple terms raised to positive powers ex) x^3 + 2x + 1), then the number of solutions to that polynomial is equal to the degree (or highest exponent) in the polynomial. The factorization in the beginning was dealing with a quadratic equation - when foiled out it equals x^2 - hx - kx + hk. The highest exponent in the quadratic is two and therefore there are two solutions. You can even think back to the factorization again: if x = h then the whole equation is 0, if x = k then the whole equation is 0.

There is no limit to what may be used as an exponent. Even infinity can be an exponent (though some things, like 1∞ , are not defined at all, and others are themselves infinite).

No. Even a number with an exponent of zero equals one. There is no way an exponent on a number will make it zero.

That's true if the exponent is zero. Then it doesn't even matter what the base is.

A polynomial of order 3 (a cubic) or higher can have more than three terms. However, the the following polynomial, even though of order 7, has only 2 terms: x7 - 23.