Factorization of Polynomial 2x² +4x

Objective

To factorize a polynomial 2x²+4x

What is a Polynomial?

  • An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
  • Expressions that contain exactly one, two, and three terms are called monomials, binomials, and trinomials respectively. In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative integers as exponents) is called a polynomial.
  • An identity is an equality, which is true for all values of the variables in the equality. On the other hand, an equation is true only for certain values of its variables. An equation is not an identity.
  • The following are the standard identities:
  • (a + b)² = a² + 2ab + b²
  • (a - b)² = a² - 2ab + b²

What is Factorization?

  • Factoring a polynomial is the process of decomposing a polynomial into a product of two or more polynomials. A polynomial can be written as the product of its factors having a degree less than or equal to the original polynomial. The process of factoring is called factorization of polynomials.

            Methods of Factorization

 

1. By splitting the middle term =>The method of Splitting the middle term is where you split the middle term into two factors.  We know that composite numbers can be expressed as the product of prime numbers. For example, 42=2×3×7. Here, 2,3, and 7 are the prime factors of 42.                               

  • This method is also said to be factoring by pairs. Here, the given polynomial is distributed in pairs or grouped in pairs to find the zeros. Let us take an example.
  • Example: Factorise x²-15x+50
  • Find the two numbers which when added gives -15 and when multiplied gives 50.
  • So, -5 and -10 are the two numbers, such that;
  • (-5) + (-10) = -15
  • (-5) x (-10) = 50
  • Hence, we can write the given polynomial as;
  • x²-5x-10x+50
  • x(x-5)-10(x-5)
  • Taking x – 5 as common factor we get;
  • (x-5)(x-10)
  • Hence, the factors are (x – 5) and (x – 10).

 

2. By using factor theorem=> According to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number,          then, (x-a) is a factor of f(x), if f(a)=0. Factor theorem is commonly used for factoring a polynomial and finding the roots of              the polynomial. It is a special case of a polynomial remainder theorem.

  • For a polynomial p(x) of degree greater than or equal to one,
  • x-a is a factor of p(x), if p(a) = 0
  • If p(a) = 0, then x-a is a factor of p(x)
  • Where ‘a’ is a real number.
  • Example: Check whether x+3 is a factor of x³ + 3x² + 5x +15.
  • Let x + 3= 0
  • x = -3
  • Now, p(x) = x³ + 3x² + 5x +15
  • Let us check the value of this polynomial for x = -3.
  • p(-3) = (-3)³ + 3 (-3)² + 5(-3) + 15 = -27 + 27 – 15 + 15 = 0
  • As, p(-3) = 0, x+3 is a factor of x³ + 3x² + 5x +15.

Area of Rectangle:

Let us say a rectangle ABCD of length=x breadth=1, then the area of rectangle ABCD can be defined as=x.

Area of Square:

Let us say a square ABCD of side=x, then the area of square ABCD can be defined as=x².