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povatix
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Homework Statement
Is it there a method to find out if a polynomial has no integer roots?
The Attempt at a Solution
I tried the division of polynomials, as well as the Horner's Method, but no luck.
povatix said:Homework Statement
Is it there a method to find out if a polynomial has no integer roots?
The Attempt at a Solution
I tried the division of polynomials, as well as the Horner's Method, but no luck.
abelian jeff said:Povatix,
Does the polynomial itself have integer coefficients? If so, you can use Eisenstein's criterion.
Polynomials are mathematical expressions that consist of variables, coefficients, and exponents. They can be written in the form of ax^n + bx^(n-1) + ... + c, where a, b, and c are constants and n is a non-negative integer.
No, polynomials do not always have integer roots. Some polynomials, such as x^2 + 1, do not have any real roots, let alone integer roots.
A polynomial with integer roots will have factors that are also integers. Therefore, you can use the Rational Root Theorem to determine potential integer roots by finding all possible factors of the constant term and dividing them by all possible factors of the leading coefficient. If any of these potential roots evaluate to 0 when substituted into the polynomial, then they are integer roots.
If a polynomial has complex roots, it means that the roots are not real numbers but rather involve the imaginary unit i. Complex roots come in conjugate pairs, so if a + bi is a root, then a - bi is also a root. In this case, the polynomial does not have any integer roots.
If a polynomial has repeated roots, it means that one or more of the roots have a multiplicity greater than 1. This means that the polynomial touches or crosses the x-axis at that root, rather than crossing through it. In this case, the polynomial may have integer roots, but not necessarily distinct integer roots.