Introduction to Factoring Polynomials

Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.

Learning Objective

Describe what it means to factor a polynomial and why it is useful to do so

Key Points

Factorization or factoring is the decomposition of an object, for example an integer or a polynomial, into a product of other objects, or factors, which when multiplied together give the original.
Factoring by grouping is done by placing the terms in the polynomial into two or more groups, where each group can be factored separately. The results of these factorizations can sometimes be combined to make an even more simplified expression.

Terms

polynomial
an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as $a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0$. Importantly, because all exponents are positive, it is impossible to divide by x.
greatest common divisor
The greatest common divisor of a set is the largest positive integer or polynomial that divides each of the numbers in the set without remainder.
factorization
An expression listing items that, when multiplied together, will produce a desired quantity.

Example

For example, to factor the polynomial $4x^2 + 20x + 3yx + 15y$, group similar terms, $(4x^2 + 20x) + (3yx + 15y)$. Factor out the greatest common factor, $4x(x+5) + 3y(x+5)$. Factor out the binomial $(x+5)(4x+3y)$.

Full Text

A polynomial consists of a sum of monomials. However, sometimes it will be more useful to write a polynomial as a product of other polynomials with smaller degree, for example to study its zeros. The process of rewriting a polynomial as a product is called factoring.

Factoring and Expanding Polynomials

Factoring is the decomposition of an algebraic object, for example an integer or a polynomial, into a product of other objects, or factors, which when multiplied together give the original. As an example, the integer $15$ factors as $3 \cdot 5$, and the polynomial $x^3 + 2x^2$ factors as $x^2(x+2)$. In all cases, a product of simpler objects than the original (smaller integers, or polynomials of smaller degree) is obtained.

For example:

$\begin{aligned}3x^3-2x^2-3xy^2+2y^2 &= (3x-2)x^2 + (2-3x)y^2 \\ & = (3x-2)(x^2-y^2) \\ & = (3x-2)(x+y)(x-y) \end{aligned}$

is a factorization of a polynomial of degree $3$ into $3$ polynomials of degree $1$.

The aim of factoring is to reduce objects to "basic building blocks", such as integers to prime numbers, or polynomials to irreducible polynomials. (These are polynomials which cannot be factored non-trivially.)

The inverse procedure of polynomial factorization is expansion, which is just explicitly writing out the multiplication of two or more factors, for example:

$(x^3-2x+5y)(2x^4-3) = 2x^7 - 4x^5 +10x^4y - 3x^3+ 6x-15y $

Example: Factoring by Grouping

One way to factor polynomials is factoring by grouping. This is done by grouping the terms in the polynomial into two or more groups in such a way that each group can be factored separately. The results of these factorizations can sometimes be combined to make an even more simplified expression. For example, to factor the polynomial $4x^2+20x+3yx+15y$, we can factor the terms with $y$ and those without $y$ separately:

$(4x^2+20x)+(3yx+15y)$

As both terms in the left expression are divisible by $4x$ and both terms in the right expression are divisible by $3y$ we can again rewrite this as:

$4x(x+5)+3y(x+5)$

Both groups share the same factor $(x+5)$, so the polynomial is factored as:

$(x+5)(4x+3y)$

Sometimes, when factoring a polynomial in two or more variables, this last step is not possible and we have to content ourselves with having two or more terms which are each factorized themselves:

$\begin{aligned} x^2 + 2x + y^2 - 2y + 2 & = (x^2 + 2x + 1) + (y^2 - 2y + 1) \\ & = (x+1)^2 + (y -1)^2. \end{aligned}$

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