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The derivative of a cubic function is a quadratic function. A cubic function with real coefficients has either one or three real roots (which may not be distinct); [1] all odd-degree polynomials with real coefficients have at least one real root. The graph of a cubic function always has a single inflection point.
If <, the cubic has one real root and two non-real complex conjugate roots. This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots.
A simplified version of the LLL factorization algorithm is as follows: calculate a complex (or p-adic) root α of the polynomial () to high precision, then use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm to find an approximate linear relation between 1, α, α 2, α 3, . . . with integer coefficients, which might be an ...
If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.
The polynomial x 2 + 1 = 0 has roots ± i. Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue. The polynomial + has roots , +,, and thus can be factored as
Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots. More generally, the discriminant ...
In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials.
The roots of this polynomial are 0 and the roots of the quadratic polynomial y 2 + 2a 2 y + a 2 2 − 4a 0. If a 2 2 − 4a 0 < 0, then the product of the two roots of this polynomial is smaller than 0 and therefore it has a root greater than 0 (which happens to be −a 2 + 2 √ a 0) and we can take α as the square