Math-for-all-Grades
Factorization of Polynomials
Basic Algebra > Factorization of Polynomials
What you will learn in Factorization of Polynomials?
In arithmetic, you are familiar with factorization of integers into prime factors.
For example, 6 = 2 х 3.
6 is called multiple, while 2 and 3 are called its divisors or factors.
The process of writing 6 as product of 2 and 3 is called factorization.
Factors 2 and 3 cannot be further reduced into other factors.
Like factorization of integers in arithmetic, we have factorization of polynomials into other irreducible polynomials in algebra.
For example, x2 + 2x is a polynomial (more specifically a binomial as it contains two terms). It can be factorized into x and (x + 2).
x2 + 2x = x (x + 2).
x and x + 2 are two factors of x2 + 2x.
While x is a monomial factor, x + 2 is a binomial factor.
In this lesson, we will discuss 6 types of factorization.
Terms occurring during factorization are clearly defined.
If you wish to set off with your lesson on factorization,
then click on this link:
The Six Types of Factorization
Or, if you wish to capture a terse overview of each factorization type, then go through each of the following header-links. You can also click the header-links to take you to the page on the specific factorization type.
Type 1: Factorization into Monomials:
Example:
6x2 + 12x = 6x (x + 2)
Here, 6x is a monomial. It is the greatest common factor of 6x2 and 12x.
Type 2: Factorization by Grouping of Terms:
Example:
4x + 4y +px + py = (4 + p) x + (4 + p) y = (p + 4)(x + y)
In this type, like terms — terms that have same literal coefficients i.e., variables such as x and y are grouped together.
Factorization will consist of product of two binomial factors.
Type3: Factorization of Perfect Square Trinomials:
Example:
4x2+ 12xy + 9y2 = (2x) 2 + 2(2x)(3y) + (3y)2= (2x + 3y)2
In this type, some of the famous algebraic identities are used.
In the polynomial, two terms are perfect squares and the other term is product of 2 and the square roots of the two perfect squares.
The trinomial form is conducive to apply algebraic identities for factorization.
Type 4: Factorization of Trinomials of the form x2+ bx + c:
Example:
x2 + 5x + 6 = x2 + 2x + 3x + 6 = x(x + 2) + 3(x + 2) = (x + 2)(x + 3)
In this type, by inspection, we find two numbers whose sum is 5, the middle term and product is 6, the constant term.
Also the sign of the middle term is written for the greater of the two numbers.
Type 5: Factorization of Trinomials of the form ax2+ bx + c:
Example:
6x2 + 5x + 1 =
6x2 + 2x + 3x + 1 = 2x (3x + 1) + 1(3x + 1) = (3x + 1)(2x +1)
In this type, we first find the product of a and c.
In the example a = 6 and c = 1
Now, as in type 4, we find two numbers whose sum is 5 and product is 6.
Type 6: Factorization of Difference of Two Perfect Squares:
4x2 — 9y2 = (2x)2 — (3y)2 = (2x + 3y) (2x — 3y)
In this type, factorization is done using the algebraic identity
a2 — b2 = (a— b)(a + b)
order of writing the binomial factors does not make any difference.
You can now click the following links which take you to the pages on the specific factorization type, or click any of the 6 links above.
Type 1: Factorization into Monomials:
Type 2: Factorization by Grouping of Terms:
Type3: Factorization of Perfect Square Trinomials:
Type 4: Factorization of Trinomials of the form x2+ bx + c:
Type 5: Factorization of Trinomials of the form ax2+ bx + c:
Type 6: Factorization of Difference of Two Perfect Squares:
