Search results
Results from the WOW.Com Content Network
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 .
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.
Modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits. [3] For this purpose, even for factoring over the rational numbers and number fields , a fundamental step is a factorization of a polynomial over a finite field .
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 ...
The quadratic integer ring [] of all complex numbers of the form +, where a and b are integers, is not a UFD because 6 factors as both 2×3 and as (+) (). These truly are different factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3, 1 + − 5 {\displaystyle 1+{\sqrt {-5}}} , and 1 − − 5 {\displaystyle 1 ...
The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra ), it follows that every polynomial with real coefficients can be factored into ...
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1] In terms of a new quantity x − h {\displaystyle x-h} , this expression is a quadratic polynomial with no linear term.
The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial x 4 − 1 {\displaystyle x^{4}-1} can be factored as follows: