# Factor an expression

To multiply, use a*b and not ab nor a x b

## 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), hence x² + 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 difference x³ - 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 is 5. The factorization gives:
15 + 20 = 5(3 + 4).
• In the expression 4x + 12x², the common factor is 4x. It is factored as:
4x + 12x² = 4x(1 + 3x).
• Consider 9xy + 27x²y. Here, the common factor is 9xy, 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.