Factor an expression
Definition and Importance of Factorization
Factorization is a key process in mathematics involving rewriting polynomials as products of simpler factors. This technique is crucial for simplifying expressions and solving equations more easily.
Main Factorization Methods
Remarkable Identities
Remarkable identities are essential tools in algebra for factorization and simplification of expressions. Here are the most commonly used ones:

Square of a Binomial:
(a + b)² = a² + 2ab + b²
(a  b)² = a²  2ab + b²
.
These formulas allow for expanding the square of a sum or a difference.

Cube of a Binomial:
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a  b)³ = a³  3a²b + 3ab²  b³
.
They facilitate the manipulation of expressions raised to the third power.

Difference of Two Squares:
a²  b² = (a + b)(a  b)
.This identity is useful for factoring a difference of squares.

Sum and Difference of Two Cubes:
a³ + b³ = (a + b)(a²  ab + b²)
a³  b³ = (a  b)(a² + ab + b²)
.These formulas are used to factor the sum and difference of cubes.
These identities play a crucial role in simplifying and solving complex equations in algebra.
Examples of Factorization Using Remarkable Identities
Here are concrete examples showing how to use remarkable identities for factorization:

Square of a Binomial: For the expression
x² + 6x + 9
, we recognize the square of the binomial(x + 3)
, hencex² + 6x + 9 = (x + 3)²
. 
Cube of a Binomial: The expression
x³  27
is the difference of two cubes, and is factored as(x  3)(x² + 3x + 9)
. 
Difference of Two Squares: For
16  x²
, we apply the difference of squares, giving(4 + x)(4  x)
. 
Sum and Difference of Two Cubes: The sum
8 + x³
is factored into(2 + x)(4  2x + x²)
, while the differencex³  8
becomes(x  2)(x² + 2x + 4)
.
Common Factor
The common factor method involves extracting a term common to all elements of an expression. For example, in the expression 4x + 8, the common factor is 4, and the expression can be rewritten as 4(x + 2).
Examples of Using the Common Factor

For the expression
15 + 20
, the common factor is5
. The factorization gives:
15 + 20 = 5(3 + 4)
. 
In the expression
4x + 12x²
, the common factor is4x
. It is factored as:
4x + 12x² = 4x(1 + 3x)
. 
Consider
9xy + 27x²y
. Here, the common factor is9xy
, which leads to:
9xy + 27x²y = 9xy(1 + 3x)
.
Product of Two Conjugate Binomials
The product of two conjugate binomials is expressed as the difference of the squares of these terms. For example, (a  b)(a + b) = a²  b²
.
Examples of Factorization of the Product of Two Conjugate Binomials

For the expression
x²  9
, we apply the product of two conjugate binomials:
x²  9 = (x + 3)(x  3)
. 
Consider
25  y²
. The factorization gives:
25  y² = (5 + y)(5  y)
. 
For
4z²  16
, we use the same method:
4z²  16 = (2z + 4)(2z  4)
. 
In the expression
1  4x²
, the factorization is:
1  4x² = (1 + 2x)(1  2x)
.
Square and Cube of a Binomial
The square of a binomial, such as (x + a)², is expanded into x² + 2ax + a². Similarly, the cube of a binomial, like (x + a)³, is expanded into x³ + 3x²a + 3xa² + a³.
Examples of Factorization of Binomials Squared and Cubed
1. Factorization of x² + 6x + 9
: This expression is the perfect square of the binomial (x + 3)
, hence x² + 6x + 9 = (x + 3)²
.
2. Factorization of y²  4y + 4
: This expression is the perfect square of the binomial (y  2)
, hence y²  4y + 4 = (y  2)²
.
3. Factorization of x³ + 12x² + 48x + 64
: This expression is the perfect cube of the binomial (x + 4)
, hence x³ + 12x² + 48x + 64 = (x + 4)³
.
4. Factorization of z³  15z² + 75z  125
: This expression is the perfect cube of the binomial (z  5)
, so z³  15z² + 75z  125 = (z  5)³
.
Sum and Difference of Two Terms to the Third Power
The sum of two cubes, x³ + a³, can be factored into (x + a)(x²  xa + a²), while the difference of two cubes, x³  a³, is factored into (x  a)(x² + xa + a²).
Examples of Factorization of the Sum and Difference of Two Cubes
1. Factorization of x³ + 27
: This expression is a sum of two cubes, x³ and 3³. Its factorization is (x + 3)(x²  3x + 9)
.
2. Factorization of y³  64
: Here, we have the difference of two cubes, y³ and 4³. The factorization is (y  4)(y² + 4y + 16)
.
3. Factorization of 125 + z³
: This expression represents the sum of two cubes, 5³ and z³. It is factored into (5 + z)(25  5z + z²)
.
4. Factorization of 8a³  27b³
: This is the difference of two cubes, (2a)³ and (3b)³. The factorization is (2a  3b)(4a² + 6ab + 9b²)
.
Conclusion
Factorization is a powerful tool in mathematics, enabling the simplification of complex expressions and the efficient solving of equations.
See also
Simplify an Expression
Mathematics Calculators