Factorization of a Polynomial

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

x² – 1 = (x - 1)(x + 1)

x² – (a + b)x + ab = (x – a)(x – b)

x⁴ – a⁴ = (x – a)(x + a)(x² + a²)

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

x⁴ + a²x² = x²(x² + a²)

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

the use of memorized formulas—

x² – a² = (x – a)(x + a)

x³ – a³ = (x – a)(x² + ax + a²)

x² + 2ax + a² = (x + a)²

x³ + 3ax² + 3a²x + a³ = (x + a)³

and factoring by grouping—

x³ + ax² + a²x + a³ = (x³ + ax²) + (a²x + a³)

= x²(x + a) + a²(x + a)

= (x + a)(a² + x²)

x⁴ + a⁴ = (x⁴ + 2a²x² + a⁴) – 2a²x²

= (x² + a²)² – ()²

= (x² - + a²)(x² + + a²)

If a polynomial of degree n

p (x) = a₀ + a₁x x + a₂x² + … + a_nxⁿ

(a_n ≠ 0) has roots x₁, x₂, …, x_n, it can be factored in the following way:

p (x) = a_n(x – x₁)…(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 x³ – 6x2 + 11 x – 6. Since it has the roots x₁ = 1, x₂ = 2, and x₃ = 3, it can be factored as follows:

x³ + 6x²+ 11x – 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 x⁴+ a⁴. 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 xⁿ + y, where n is any natural number. Such a polynomial is said to be irreducible.