Factoring Polynomials

A polynomial is an algebraic expression made up of terms that include constants and variables raised to non-negative integer exponents, combined using addition, subtraction, or multiplication.


They do not include square roots, negative powers, or variables in denominators, but include operations like addition, subtraction, and multiplication.
 

Factoring a polynomial requires breaking it down into smaller factors that, when multiplied, give the original expression. This method helps solve equations and better understand the behavior of the polynomial.
A polynomial is typically represented as axⁿ + bx^n-1 +... + px + q and can be factored using grouping, substitution, or identities.


The highest power of x, called the degree, helps determine how many solutions (or zeros) the polynomial can have. Factoring changes a complicated polynomial into smaller pieces, like expressions with x or x², that are easier to work with. 


For instance, x² + x(a + b) + ab becomes (x + a)(x +b)
 

To factorize a polynomial, follow these steps:


Step 1: Look at all the terms and take out the greatest common factor (GCF), if there is one.


Step 2: Use a factoring method that fits the expression, such as grouping, using algebraic identities, or trying substitution.


Step 3: Rewrite the polynomial as a product of the smaller expressions.
 

The six different types of methods used for factorizing a polynomial are listed below:

 

1. Greatest Common Factor (GCF)
   
    
2. Grouping Method
   
    
3. The Sum of the Difference in Two Cubes
   
    
4. Difference in Two Squares Method
   
    
5. General Trinomials
   
    
6. Trinomial Method

In this section, we will focus on three main methods used to solve polynomials.

Greatest Common Factor

In this method, we take out the largest number (and variable, if any) that all the terms have in common. It’s like reversing the distributive property.


Distributive property: 


p(q + r) = pq + pr


Factored form:


pq + pr = p(q + r), where p is the greatest common factor.

 

Example: Factor the polynomial 12x + 18

 

Step 1: Find the GCF 


Break each term into its factors:


\(12x = 3 × 4 × x\) or \(6 × 2 × x\)


\(18 = 3 × 6\)


So, the GCF is 6.

 

Step 2: Factor it out


\(12x + 18 = 6(2x + 3)\)

 

Step 3: Check by distributing



\(6(2x + 3) = 6 \times 2x + 6 \times 3 = 12x + 18\)

Factoring polynomials by grouping 

This method works well while trying to factor trinomials (three-term expressions) that can’t be factored just by taking out a common factor. The idea is to split the middle term into two parts so we can group the terms and factor in pairs.

 

Example: Factor \(x^2 + 11x + 24 = (x + 3)(x + 8)\)


Step 1: Find two numbers


Look for two numbers that:


Add up to 11 (the middle term), and


Multiply by 24 (the last term)


The numbers 3 and 8 work because:


\(3 + 8 = 11\)


\(3 × 8 = 24\)

 

Step 2: Rewrite the middle term using these two numbers


\(x^2 + 11x + 24 = x^2 + 3x + 8x + 24\)


Step 3: group the terms and factor


\((x^2 + 3x) + (8x + 24)\)


\(x(x + 3) + 8(x + 3) \)

Take the common binomial factor, 


\((x + 3)(x + 8)\)


So, \((x + 3)(x + 8)\) are the factors of \(x^2 + 11x + 24\)

Factoring using Identities

Algebraic identities can also help in factorizing polynomials. Commonly used identities for this purpose are:


\((a + b)^2 = a^2 + 2ab + b^2\)


\((a - b)^2 = a^2 - 2ab + b^2\)


\(a^2 - b^2 = (a + b)(a - b)\)


Let’s take examples for each identity listed above.


Example 1: Factorize \(x^2 + 6x + 9\), using (a + b)²


Step 1 - Compare with the identity


\(x^2 + 6x + 9 = x^2 + 2 \times 3 \times x + 3^2\)


We can see the pattern a² + 2ab + b2, where a = x and b = 3


Step 2 - Apply the identity


\(x^2 + 6x + 9 = (x + 3)^2\)


So, (x + 3)² is the answer.


Example 2: Factor \(x^2 - 10x + 25\) using identity (a - b)² 


Step 1 - Match with the identity


\(x^2 - 10x + 25 = x^2 - 2 \times 5 \times x + 5^2\)


Here, a = x and b = 5


Step 2 - Apply the identity


\(x^2 - 10x + 25 = (x - 5)^2\)


(x - 5)2 is the answer.


Example 3: Factor x² - 49 using identity a² - b²


Step 1 - It needs to be the difference between two perfect squares


\(x^2 - 49 = x^2 - 7^2\)


Step 2 - Apply Identity a² - b² = (a + b)(a - b)


So, \(x^2 - 49 = (x + 7)(x - 7)\)
 

Factoring polynomials can seem tricky at first, but with the right techniques and a little practice, it becomes simple. These tips and tricks make the process faster and more accurate.

 

• Always start by factoring out the largest number or variable common to all terms. This simplifies the polynomial and makes further factoring easier.
  
   
• For quadratic trinomials like ax²+bx+c, split the middle term into two numbers whose product is ac and sum is b.
  
   
• When a polynomial has four terms, try factoring by grouping — combine terms in pairs and factor out the common parts.
  
   
• If the expression looks complex, substitute a variable (e.g., let y=x²) to make it easier to recognize a pattern.
  
   
• Multiply your factors back to ensure they give the original polynomial — a quick way to confirm accuracy.

Factoring polynomials is an important algebraic concept that helps design structures and predict profits. It has a wide range of real-life applications, and some of them are mentioned below: 

• Engineering and Design: Engineers use factoring to determine dimensions, stresses, and load distributions in mechanical and structural designs. It helps simplify motion equations and material strength calculations.
  
   
• Physics and Motion Analysis: In physics, polynomial equations often describe motion, force, and energy. Factoring helps in finding critical points like maximum height, equilibrium, or velocity changes.
  
   
• Robotics and Automation: Robotics uses polynomial equations to model motion paths and control algorithms. Factoring simplifies these equations for efficient computation and error reduction.
  
   
• Architecture: Architects use factoring to solve area, volume, and design optimization problems where polynomial equations are used to represent structural elements.