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 + bxn-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  2x + 6  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 x2 + 11x + 24
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
x2 + 11x + 24 = x2 + 3x + 8x + 24
Step 3 - group the terms and factor
(x2 + 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 x2 + 11x + 24

Factoring using Identities

Algebraic identities can also help in factorizing polynomials. Commonly used identities for this purpose are:
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
a2 - b2 = (a + b)(a - b)
Let’s take examples for each identity listed above.
Example 1: Factorize x2 + 6x + 9, using (a + b)2
Step 1 - Compare with the identity
x2 + 6x + 9 = x2 + 2  3  x + 32
We can see the pattern a2 + 2ab + b2, where a = x and b = 3
Step 2 - Apply the identity
x2 + 6x + 9 = (x + 3)2
So, (x + 3)2 is the answer.
Example 2: Factor x2 - 10x + 25 using identity (a - b)2 
Step 1 - Match with the identity
x2 - 10x + 25 = x2 - 2  5  x + 52
Here, a = x and b = 5
Step 2 - Apply the identity
x2 - 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
x2 - 49 = x2 - 72
Step 2 - Apply Identity a2 - b2 = (a + b)(a - b)
So, x2 - 49 = (x + 7)(x - 7)
 

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: 

Finding break-even points in business

Let’s say we run a business, and the profit depends on how many items we sell. If our profit equation is: 
Profit = x2 - 6x - 7
To find when we’ll make zero profit (break even), we should set the profit equation to 0. 
So, x2 - 6x - 7 = 0
Factoring it:
(x - 7)(x + 1) = 0
So, x = 7 or x = -1
Since in real life, we can’t sell -1 item, the value to be taken is 7. So, we’ll break even after selling 7 items.

Solving motion problems in physics

Let's say a ball is thrown upward, and its height over time is given by the equation:
h = -5t² + 20t
To find when the ball will hit the ground, set the equation to 0.
So, -5t² + 20t = 0
Factoring it:
-5t(t - 4) = 0
So, t = 0 or t = 4
This means the ball will hit the ground exactly 4th second after being thrown in the air.

Determining dimensions in architecture

Engineers and architects factor area and volume polynomials to find missing dimensions like length, breadth, and height in structural designs.

Optimizing surface area in design and manufacturing

When companies design boxes or cans, they want to use as little material as possible while still meeting size needs. That means they need to minimize surface area. Many surface area formulas are quadratic expressions. By factoring them, manufacturers can find the optimum length, width, or height that uses the least material, saving both cost and resources.

Modeling the spread of disease or population in biology and epidemiology

Growth models are polynomials that are factored to find turning points or critical population thresholds. For example, a population function t2 - 5t + 6 gives us the times of growth change when factored.