Factored Form

The process of dividing a number or algebraic expression to its relevant factors is known as factoring. When you multiply these factors, it should give back the original expression.

For example: 
x2 – 7x + 12

We have to find two numbers whose product is 12 and sum is -7.

We have the numbers: -3 and -4.

So,
x² – 7x + 12 = (x - 3) (x - 4)

Verify the factoring by multiplying the factors together. 

(x - 3) (x - 4) = x² – 4x -3x + 12 = x² - 7x + 12.

Since we get back the original expression, the factorization is correct.

Factoring plays a major role in algebra and is done through a series of steps. Let’s go through each steps in detail:

The largest expression in an algebraic expression is the Greatest Common Factor(GCF), which includes both the variables and numerical coefficients; they can divide each term exactly without leaving a remainder.  
 
For example:
Factor: 12x²y – 8xy² + 16xy

Solution:

To simplify an algebraic expression, we first need to simplify its GCF.

GCF of all terms = 4xy

Let’s now factor it out:

4xy(3x – 2y + 4)

Factoring by grouping works well for polynomials that have four terms. In this the terms are grouped into two pairs, and the common factor is taken out from each group. 

For example:

Factor: x³ + 2x² + 3x + 6

Solution:

Group the terms:

(x³ + 2x²) + (3x + 6)

Take out the common factor from each group of terms.

x²(x + 2) + 3(x + 2)

Now, we factor out further:

(x + 2)(x² + 3) 

Here, x2 + 3 cannot be factorized more.

A trinomial is an algebraic expression made up of three terms. Let’s explore how to factor a quadratic expression using the AC method. 

For example: 

Factor: x² + 7x + 10

Solution:

Find two numbers whose product is 10 and sum is 7.

2 and 5

Rewrite the middle term using the factors:

x² + 2x + 5x + 10

We now group the terms into pairs:

(x² + 2x) + (5x + 10)

Factor out each group:

x(x + 2) + 5(x + 2)

Factor out further for the common binomial:

(x + 2)(x + 5)

This method can be used only in cases where both terms are perfect squares separated by a minus sign. 

Formula:

a² - b² = (a + b)(a - b)

For example: 

Factor: x² - 49

Solution:

Rewrite it as a difference of squares:

x² - 7² = (x + 7)(x - 7)

An expression with three teams is called a perfect square trinomial. A perfect square trinomial will match any one of the following forms:

a² + 2ab + b² = (a + b)²

a² - 2ab + b² = (a - b)²

Example:

Factor: x² + 8x + 16

Solution:

Identify the pattern:

x² + 2·4·x + 4² = (x + 4)²

These expressions follow the specific formulas which are given below: 

a³ + b³ = (a + b)(a² - ab + b²)

a³ - b³ = (a - b)(a² + ab + b²)

For example: 

Factor: x³ + 64

Solution:

Identify it as a sum of cubes:

x³ + 64 = x³ + 4³ 

The formula we use: a³ + b³ = (a + b)(a² - ab + b²)

Applying the formula:

x³ + 4³ = (x + 4)(x² - 4x + 16)

This method applies to factor higher-degree polynomials. Here, we substitute a repeated power with a variable.

For example: 

Factor: x⁴ + 2x² - 15

Solution:

Let y = x²

So the expression becomes:

y² + 2y - 15

Now, write the expression as a multiplication of its factors:

(y + 5)(y - 3)

Here, we substitute x² back for y:

(x² + 5)(x² - 3)

Factored form represents an expression as a product of its factors. This method can be a little tricky for some students. We will now go through some simple tricks to help you master the concept effectively.

• Check if all the terms in the expression share a greatest common factor. Ensure that you first take that common factor out.
  For example:
  
  10x² + 5x = 5x(2x + 1)

• Be attentive when it comes to the signs while factoring. It helps you check that the factoring is correct.

• To factor out the expressions, it is important to learn the common patterns, such as:
  
  a²  – b² = (a + b) (a – b)
  
  a² + 2ab + b2 = (a + b)²
  
  a² – 2ab + b2 = (a – b)²

• Reverse checking can be used to ensure that the factoring is correct.
  
  For example: 
  
  (x – 5) (x + 2)
  
  Expanding the factors gives back the original expression:
  
  x² – 3x – 10

The factored form is a useful method for representing algebraic expressions as a product of their factors. This concept is not confined to mathematics; it has widespread practical applications in real life.  Let’s now learn how it can be applied in real-world situations.

• In sports, especially in games like football or basketball, factoring quadratic equations can help predict the ball’s path, including where it will hit the ground.  

• This concept can be applied to determine the time it takes to reach the destination when traveling.

• In finance, factored form can be used to solve quadratic equations that model compound interest or investment growth, helping to determine the time or rate needed to reach a specific financial goal.