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Descartes' rule of signs will not necessarily tell exact number of complex roots, but will give an idea. The Wikipedia article explains it pretty well, but here is a brief explanation:

It is for single variable polynomials.

- Order the polynomial in descending powers [example: f(x) = axÂ³ + bxÂ² + cx + d]
- Count number of sign changes between consecutive coefficients. This is the maximum possible real positive roots.
- Let function g(x) = f(-x), count number of sign changes, which is maximum number of real negative roots.

Note these are maximums, not the actual numbers. Let p = positive maximum and q = negative maximum. Let m be the order (maximum power of the variable), which is also the total number of roots.

So m - p - q = minimum number of complex roots. Note complex roots always occur in pairs, so number of complex roots will be {0, 2, 4, etc}.

Q: How could you use Descartes' rule to predict the number of complex roots to a polynomial?

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In answering this question it is important that the roots are counted along with their multiplicity. Thus a double root is counted as two roots, and so on. The degree of a polynomial is exactly the same as the number of roots that it has in the complex field. If the polynomial has real coefficients, then a polynomial with an odd degree has an odd number of roots up to the degree, while a polynomial of even degree has an even number of roots up to the degree. The difference between the degree and the number of roots is the number of complex roots which come as complex conjugate pairs.

Descartes' rule of signs (see related link) can help you determine the maximum number of real roots. If the polynomial is odd powered, then there will be at least one real root. Any even powered polynomial can be factored into a bunch of quadratics [though they may not be rational or even pretty], and any odd-powered polynomial can be factored into a bunch of quadratics and one linear (this one would have the real root). So the quadratics may have pairs of real or complex roots (having an imaginary component).To clarify, when I say complex, I'm referring to the fact that there will be an imaginary component to the root, because actually the real numbers is a subset of the set of complex numbers.The order of the polynomial will tell you how many roots it will have. If you can graph the polynomial, then you can see if it crosses the x axis. If it is a 5th order polynomial, and crosses the x axis 3 times, then there are 3 real roots (the other two roots are complex).

An algebraic number is one which is a root of a non-constant polynomial equation with rational coefficients. A transcendental number is not an algebraic number. Although a transcendental number may be complex, Pi is not.

13 is not a polynomial.

A root.

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An algebraic number is a complex number which is the root of a polynomial equation with rational coefficients.

In answering this question it is important that the roots are counted along with their multiplicity. Thus a double root is counted as two roots, and so on. The degree of a polynomial is exactly the same as the number of roots that it has in the complex field. If the polynomial has real coefficients, then a polynomial with an odd degree has an odd number of roots up to the degree, while a polynomial of even degree has an even number of roots up to the degree. The difference between the degree and the number of roots is the number of complex roots which come as complex conjugate pairs.

Descartes' rule of signs (see related link) can help you determine the maximum number of real roots. If the polynomial is odd powered, then there will be at least one real root. Any even powered polynomial can be factored into a bunch of quadratics [though they may not be rational or even pretty], and any odd-powered polynomial can be factored into a bunch of quadratics and one linear (this one would have the real root). So the quadratics may have pairs of real or complex roots (having an imaginary component).To clarify, when I say complex, I'm referring to the fact that there will be an imaginary component to the root, because actually the real numbers is a subset of the set of complex numbers.The order of the polynomial will tell you how many roots it will have. If you can graph the polynomial, then you can see if it crosses the x axis. If it is a 5th order polynomial, and crosses the x axis 3 times, then there are 3 real roots (the other two roots are complex).

No. Complex zeros always come in conjugate pairs. So if a+bi is one zero, then a-bi is also a zero.The fundamental theorem of algebra says"Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers."If you want to know how many complex root a given polynomial has, you might consider finding out how many real roots it has. This can be done with Descartes Rules of signsThe maximum number of positive real roots can be found by counting the number of sign changes in f(x). The actual number of positive real roots may be the maximum, or the maximum decreased by a multiple of two.The maximum number of negative real roots can be found by counting the number of sign changes in f(-x). The actual number of negative real roots may be the maximum, or the maximum decreased by a multiple of two.Complex roots always come in pairs. That's why the number of positive or number of negative roots must decrease by two. Using the two rules for positive and negative signs along with the fact that complex roots come in pairs, you can determine the number of complex roots.

No.A polynomial is not a number. Do you mean can every number be represented by a polynomial?If so, the answer is still no.

There cannot be such a polynomial. If a polynomial has rational coefficients, then any complex roots must come in conjugate pairs. In this case the conjugate for 2-3i is not a root. Consequently, either (a) the function is not a polynomial, or (b) it does not have rational coefficients, or (c) 2 - 3i is not a root (nor any other complex number), or (d) there are other roots that have not been mentioned. In the last case, the polynomial could have any number of additional (unlisted) roots and is therefore indeterminate.

Is also a number or polynomial.

An algebraic number is a number that is a root of a non-zero polynomial with rational coefficients. A transcendental number is a real or complex number that is not an algebraic number. Two notable examples are pi and e.

Given ANY number it is possible to find a polynomial of order 6 such that it will predict it as the next number in the pattern. The position to value rule:Un= (2n5- 37n4+ 260n3-851n2+ 1274n + 624)/24 for n = 1, 2, 3, ... predicts 23.Given ANY number it is possible to find a polynomial of order 6 such that it will predict it as the next number in the pattern. The position to value rule:Un= (2n5- 37n4+ 260n3-851n2+ 1274n + 624)/24 for n = 1, 2, 3, ... predicts 23.Given ANY number it is possible to find a polynomial of order 6 such that it will predict it as the next number in the pattern. The position to value rule:Un= (2n5- 37n4+ 260n3-851n2+ 1274n + 624)/24 for n = 1, 2, 3, ... predicts 23.Given ANY number it is possible to find a polynomial of order 6 such that it will predict it as the next number in the pattern. The position to value rule:Un= (2n5- 37n4+ 260n3-851n2+ 1274n + 624)/24 for n = 1, 2, 3, ... predicts 23.

a polynomial of degree...............is called a cubic polynomial

no...

An algebraic number is one which is a root of a non-constant polynomial equation with rational coefficients. A transcendental number is not an algebraic number. Although a transcendental number may be complex, Pi is not.