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An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents, [14] which is an early part of Galois theory. [15] This method can be generalized to give the roots of cubic polynomials and quartic polynomials , and leads to Galois theory, which allows one to understand the solution of algebraic equations of any ...
Given a quadratic polynomial of the form + + it is possible to factor out the coefficient a, and then complete the square for the resulting monic polynomial. Example: + + = [+ +] = [(+) +] = (+) + = (+) + This process of factoring out the coefficient a can further be simplified by only factorising it out of the first 2 terms.
For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. [ 6 ] : 202–207 If one is given a quadratic equation in the form x 2 + bx + c = 0 , the sought factorization has the form ( x + q )( x + s ) , and one has to find two numbers q and s that add up to b and whose product is c ...
Otherwise, the quadratic polynomial cannot be factorized into non-constant real factors. The quadratic formula is valid when the coefficients belong to any field of characteristic different from two, and, in particular, for coefficients in a finite field with an odd number of elements. [9]
If two or more factors of a polynomial are identical, then the polynomial is a multiple of the square of this factor. The multiple factor is also a factor of the polynomial's derivative (with respect to any of the variables, if several). For univariate polynomials, multiple factors are equivalent to multiple roots (over a suitable extension field).
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If the given polynomial only has real coefficients, one may wish to avoid computations with complex numbers. To that effect, one has to find quadratic factors for pairs of conjugate complex roots. The application of the multidimensional Newton's method to this task results in Bairstow's method.
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