Factorization: Polynomials

Factorization is a method of breaking down complex mathematical objects into simpler units (factors) that may be multiplied together to produce the original object. It is commonly used to simplify polynomials across many subdivisions of mathematics and science.

An nth degree polynomial (or polynomial of degree n) in one variable is an algebraic expression of the form

where is a natural number, and are complex numbers, and . A defining characteristic of a polynomial is that whenever the variable appears, it is raised only to a non-negative integer power. It is never raised to a fractional or negative power. For example and 1/2x⁴ + 10x³ are polynomials in x of degree 2, 3, and , respectively. An nth degree polynomial that has one term or two terms is called a monomial or binomial, respectively. For example 10x³ is a monomial in , is a binomial in . A monomial is also known as a term. A 1st degree polynomial is called a linear polynomial. For example is a linear monomial in , and is a linear binomial in . A polynomial of degree 0 is called a constant polynomial, so named because of the absence of variables such an expression. For example –6 is a constant polynomial.

Polynomials need not be in only one variable. An algebraic expression in two variables in which each variable is only ever raised to a non-negative integer power is also considered a polynomial. The degree of each term is the sum of the exponents on the variables, and the degree of the polynomial is the degree of the highest degree term. For example is a polynomial term of degree 2 + 3 = 5 in the variables and , and is a polynomial of degree 7 in the variables and .

Factorization (or simply factoring) is an extremely important algebraic skill to possess for a variety of reasons, not least of which is being able to find solutions to polynomial equations. Factoring may be considered the reverse process of distributing one polynomial over another, as illustrated in the example below.

It is perhaps not surprising then that each rule for the factorization of polynomials is based on a form of the distributive law. A polynomial is considered completely factored if none of its factors is divisible by any other polynomial (including constant polynomials). Specific factorizations of polynomials include factoring linear binomials, factoring polynomials by grouping, and factoring polynomials with two variables.

Greatest common factor (GCF) refers to a set of polynomial terms and how to find it. The GCF of a set of polynomial terms is the term of highest possible degree and coefficient that is a divisor of each term in the set. To find the GCF of a set of terms, find both the coefficient and the exponents on the variables. The coefficient of the GCF is the GCF of the coefficients of the terms, and the exponent of each variable in the GCF is the smallest exponent found on each corresponding variable in the set of terms. For example if the set of terms is and , then the GCF is .

Factoring Linear Binomials

Each factoring rule is based on a form of the distributive law, in this case:

where is the GCF of the two terms on the left side of . When looked at in reverse, the distributive law becomes a factoring rule:

Example 1:

Both terms in Example 1 have positive coefficients. That need not always be the case. One or both terms could have negative coefficients, but the factorization process is still the same, as Examples 2 and 3 illustrate.

Example 2:

Example 3:

Because factoring is the reverse of distributing, it is always possible to check one's factoring work by distributing. For instance in Example 3, .

Factoring Polynomials by Grouping

Factoring by grouping is based on , except that when factoring by grouping the GCF can be more than just a monomial. In an algebra course, factoring by grouping is the method that one usually tries to use when factoring a polynomial of four or more terms.

Example 4:

The GCF of the four terms in an expression need not be 1 in order to be factorable by grouping. If the GCF is something other than 1, then it should be factored out before factoring by grouping. But how should a GCF be factored out of an expression with four terms? The factoring rule is a generalization of (2) to four terms, namely:

generalizes to any number of terms in an analogous way.

Example 5:

Finally, it is entirely possible that other factorization rules may have to be used in conjunction with factoring by grouping, as Examples 6 and 7 illustrate.

Example 6:

Factoring Polynomials with Two Variables

Linear binomials need not have only one variable. For instance is a linear binomial in two variables, and it can also be factored using .

Example 8:

Further, it need not be the case that linear binomials have only two terms. For instance is a linear polynomial with four terms, and it is factorable. The GCF of the four terms is .

Example 9:

Also, an expression need not have just one variable to be factorable by grouping, as Example 10 demonstrates.

Example 10:

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