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 integers by excluding the fractional factors. Then the rational zeros are determined and the polynomial is decomposed into its corresponding linear factors. Factors that do not have rational zeros are not decomposed further.
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
Irrational zeros : -0,57735, 0,57735
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 Sketching.
See also:
Wikipedia: Factorization of polynomials