# Factorization of a Polynomial

## Factorization of a Polynomial

the representation of a polynomial as a product of two or more polynomials of lower degrees. For example,

*x*^{2} – 1 = (*x* - 1)(*x* + 1)

*x*^{2} – (*a* + *b*)*x* + *ab* = (*x* – *a*)(*x* – *b*)

*x*^{4} – *a*^{4} = (*x* – *a*)(*x* + *a*)(*x*^{2} + *a*^{2})

The simplest methods of factoring include the removal of a common factor—

*x*^{4} + *a*^{2}*x*^{2} = *x*^{2}(*x*^{2} + *a*^{2})

*x* (*x* – *a*) – *b* (*x* – *a*) = (*x* – *a*)(*x* – *b*)

the use of memorized formulas—

*x*^{2} – *a*^{2} = (*x* – *a*)(*x* + *a*)

*x*^{3} – *a*^{3} = (*x* – a)(*x*^{2} + *ax* + *a*^{2})

*x*^{2} + 2*ax* + *a*^{2} = (*x* + *a*)^{2}

*x*^{3} + 3*ax*^{2} + 3*a*^{2}*x* + *a*^{3} = (*x* + *a*)^{3}

and factoring by grouping—

*x*^{3} + *ax*^{2} + *a*^{2}*x* + *a*^{3} = (*x*^{3} + *ax*^{2}) + (*a*^{2}*x* + *a*^{3})

= *x*^{2}(*x* + *a*) + *a*^{2}(*x* + *a*)

= (*x* + *a*)(*a*^{2} + *x*^{2})

*x*^{4} + *a*^{4} = (*x*^{4} + 2*a*^{2}*x*^{2} + *a*^{4}) – 2*a*^{2}*x*^{2}

= (*x*^{2} + *a*^{2})^{2} – ()^{2}

= (*x*^{2} - + *a*^{2})(*x*^{2} + + *a*^{2})

If a polynomial of degree *n*

*p* (*x*) = *a*_{0} + *a*_{1}*x x* + *a*_{2}x^{2} + … + *a*_{n}*x ^{n}*

(a_{n} ≠ 0) has roots *x*_{1}, *x*_{2}, …, *x*_{n}, it can be factored in the following way:

*p* (*x*) = *a*_{n}(*x* – *x*_{1})…(*x* – *x*_{n})

Here, all the factors are linear—that is, they are of the first degree. As an example, let us take the cubic polynomial *x ^{3} – 6x2* + 11

*x*– 6. Since it has the roots

*x*

_{1}= 1,

*x*

_{2}= 2, and

*x*

_{3}= 3, it can be factored as follows:

*x*^{3} + 6*x*^{2}+ 11*x* – 6 = (*x* – 1)(*x* – 2)(*x* – 3)

In general, every polynomial with real coefficients can be resolved into linear and quadratic factors that also have real coefficients. An example is the factorization given above for *x ^{4}+ a^{4}.* In this case, both factors are quadratic. If

*a*is real and nonzero, the two factors can be themselves resolved only into factors with complex coefficients; for example,

There exist polynomials in two or more variables of arbitrarily high degree that cannot be factored at all. An example is the polynomial *x ^{n} + y,* where

*n*is any natural number. Such a polynomial is said to be irreducible.

### REFERENCE

Kurosh, A. G.*Kurs vysshei algebry,*10th ed. Moscow, 1971.

A. I. MARKUSHEVICH