Factorization of Polynomials

The rational zeros and the linear factor decomposition of a polynomial are determined.

p(x) = a9·x9 + a8·x8 + ... + a0.

The coefficients of the polynomial can be entered as fractions, as mixed numbers or as breaking decimal numbers.

p(x) = x5 - 9·x4 - 82/9·x3 + 82·x2 + x - 9
       = (1/9)·(9·x5 - 81·x4 - 82·x3 + 738·x2 + 9·x - 81)                       
       = (1/9)·(3·x - 1)·(3·x + 1)·(x - 9)·(x - 3)·(x + 3)

Rational zeros: 1/3, -1/3, 9, 3, -3

First, the coefficients are brought to whole numbers by excluding the fraction factors. The rational zeros are then determined and the polynomial broken down into the associated linear factors. Factors that have no rational zeros are not further broken down.

Further examples:

p(x) = x6 + x5 - 5·x4 - 5·x3 + 4·x2 + 4·x                                             
       = x·(x - 1)·(x + 1)2·(x - 2)·(x + 2)
	   
Rational zeros: 0, 1, -1, 2, -2
p(x) = x6 - 36·x5 + 505·x4 - 3480·x3 + 12139·x2 - 19524·x + 10395
      = (x - 1)·(x - 3)·(x - 5)·(x - 7)·(x - 9)·(x - 11)

Rational zeros: 1, 3, 5, 7, 9, 11
p(x) = 0,2·x5 + x4 + 2·x3 + 2·x2 + x + 0,2                                         
     = (1/5)·(x5 + 5·x4 + 10·x3 + 10·x2 + 5·x + 1)       
     = (1/5)·(x + 1)5

Rational zeros: -1
p(x) =  -432·x5 - 648·x4 + 837·x3 + 1835·x2 + 875·x + 125             
      = (3·x + 1)2·(3·x - 5)·(4·x + 5)2

Rational zeros: -1/3, 5/3, -5/4
p(x) = x5 + 3·x4 + 8/3·x3 - x - 1/3                                                     
       = (1/3)·(3·x5 + 9·x4 + 8·x3 - 3·x - 1)
       = (1/3)·(x + 1)3·(3·x2 - 1)

Rational zeros: -1

If, as in the last example, a residual polynomial with a degree less than or equal to 4 remains, any remaining irrational zeros   can be determined using the program part Algebra/Equations of 4th degree.

If the degree of the residual polynomial is greater than 4, it is still possible   to search graphically for further zeros using the program part Analysis/Curve Discussion.

See also:

Wikipedia: Factorization of polynomials