Factors of a Polynomial

The opposite of multiplying factors of polynomials is factoring the polynomial. A polynomial of degree “n” in variable x is an expression of the form axn+bxn-1+kcxn-2+ ... +kx+i, where each variable has a constant associated with it as its coefficient. Therefore, a polynomial is an expression made up of one or more terms, where each term has a constant multiplied by a variable to a whole-number power, and the terms are separated by addition or subtraction.

What is Factorization of a Polynomial?

Factorization is the process of determining the numbers or expressions whose product is the original value. 
Likewise, when it comes to polynomials, the factors are the polynomials that are multiplied to create the original polynomial. The factors of x2+5x+6 are, for example, (x + 2) (x + 3). The original polynomial is produced when multiplying them. We can also determine the polynomial’s zeros after factorization. The zeros are x = -2 and x = -3.
 

Polynomials can be factorized using six techniques.

• Greatest Common Factor (GCF)
• Grouping Method
• Sum or Difference in Two Cubes
• Difference in Two Squares Method
• General Trinomials
• Trinomial Method

How to Solve Polynomials?

Polynomials can be factorized in several ways. Let’s talk about these techniques.
Greatest Common Factor
To factorize a polynomial, first find the greatest common factor of the polynomial. This procedure is simply the distributive law applied in reverse, like
p(q + r)=pq + pr
However, factorization is simply the opposite process;
pq+pr=p(q+r)
Where p is the greatest common factor.

Factoring Polynomials by Grouping

Another name for this method is factoring by pairs. Here, the zeros are found by distributing or grouping the given polynomial in pairs. Let us look at an example.
Factorize x2-15x+50
Now, find the two numbers which, when added, give -15 and when multiplied, give 50.
-5 and -10 are the two numbers such that,
(-5)+(-10)=-15
(-5) × (-10)=50
Now, the polynomial is as follows,
x2-5x-10x+50
x(x-5)-10(x-5)
Let us take x-5 as a common factor, and then we get:
(x-5)(x-10)
Therefore, the factors are (x-5) and (x-10).

Factoring Polynomial with Four Terms

Let us now see how polynomials with four terms are factorized. For example, the polynomial is x3+x2-x-1
Now, separate the given polynomial into two.
(x3+x2)+(-x-1)
Now find the highest common factor and take that factor out of the bracket.
Here x2 is the greatest common factor, and from the second part, we can factor out -1. Thus,
x2(x+1)-1(x+1)
x2(x+1)-1(x+1)=(x2-1)(x+1)
=(x-1)(x+1)(x+1)
=(x-1)(x+1)2
Now, group the terms as factors.
(x2-1)(x+1)
Thus, the factorization of  x3+x2-x-1 gives (x2-1)(x+1)
 

This shows how factors help in solving problems in real-world problems. Let us see some applications of the factors of polynomials.

• Calculating satellite orbits
  Satellite motion is modeled by polynomial equations. To determine important details like takeoff angles or re-entry routes, engineers factor these polynomials. Proper launch, orbit, and return are guaranteed by accurate factoring, which reduces risks and assures mission success.
• Analysis of mechanical vibrations
  Natural frequencies of machinery are described by polynomial equations. Resonance points can be found by factoring their characteristic polynomials. Engineers can design dampers or modify materials to prevent damaging vibrations by identifying the roots.
• Command system stability
  Characteristic polynomials govern the behavior of the control system. Considering these factors reveals pole locations, ensuring system stability in manufacturing, automotive, and aerospace controls.
• Filtering digital signals
  Polynomial transfer functions are used in filter design. By isolating zeros and poles, factoring the numerator and denominator polynomials allows for precise control over frequency response and the elimination of undesired noise.
• Planning the root’s path
  Polynomial equations are often used to define a robotic path. In automated manufacturing and autonomous vehicles, factoring these equations helps identify key waypoints that ensure smooth and collision-free motion.