Techniques for Factoring Polynomials

Common Factors 

The first method is the common factors method, in which if there is a common factor for each term in the polynomial, we factor that common term out and write the remaining polynomial.

The first approach is known as the “common factors method,” in which the common factors of each term are factored out in the polynomial.

Example: What are the factors of 3x² + 6x +12?

Solution:

Let f(x) = 3x² + 6x +12

As 3 is present in each term as expression can be rewritten as

f(x) =3x²+3×2x-3×4

⇒ f(x) = 3(x²+2x+4)

Thus, factors of the 3x² + 6x +12 are 3 and x² + 2x +4.

Grouping Method 

This method factors polynomials by grouping terms with two or more terms together and finding the greatest common factor for each grouping. Once the common factors for each grouping were found, each group had the same factor.

Example: Factorize the expression ax²+7abx+ax+7ab as a and b are some real numbers.

Solution:

Let f(x) = ax²+7abx+ax+7ab

Grouping two elements at a time,

⇒ f(x) = (ax²+ax)+(7abx+7ab)

⇒ f(x) = ax(x+1)+7ab(x+1)

⇒ f(x) = (ax+7ab)(x+1)

Thus, ax+7ab and x + 1 are the required factors.

Splitting the Middle Term 

Using this technique; the quadratic polynomials with a leading coefficient of 1, are factorized. This method is highly useful since higher-degree polynomials are frequently converted to quadratic polynomials using the factor theorem.

• Step 1: f(x) = x²+(a+b)x+ab 
• Step 2: f(x)= x²+ax+bx+ab 
• Step 3: f(x)=  x(x+a)+b(x+a) 
• Step 4: f(x)= (x+a)(x+b)

So all quadratic polynomial of form x²+(a+b)x+ab, can be factorize using Splitting the Middle Term method,

Example: Factorize x²+5x+6.

Solution:

Let f(x) = x²+5x+6

f(x)= x² + (2+3)x+2×3 

⇒ f(x)= x²+2x+3x+2×3 

⇒ f(x)= x(x+2)+3(x+2) 

⇒ f(x)= (x+2)(x+3)  

Thus, x+2 and x+3 are the factors of the expression x²+5x+6.  

Factoring Polynomials: A basic algebraic concept called factoring polynomials involves breaking down a polynomial equation into simpler parts. Factoring can be used to solve equations, simplify complicated expressions, and locate the roots or zeros of polynomial functions. 

In several fields of mathematics, including engineering, physics, and computer science, the ability to factor is a crucial skill. Finding the common factors, or roots, of the equation and breaking them down into a set of simpler expressions are the general steps involved in factoring a polynomial. 

In this article, we have provided details about factoring polynomials, steps to factorize polynomials, with solved examples and practice problems on it.