Factorization of polynomials

Polynomial: Let x be a variable (literal), n be a positive integer and a˳ , a₁ , a₂ , a₃ ......an be constants (real numbers). Then an xᶯ  + an-1. xᶯ⁻¹ + an-2. xᶯ⁻² + a₁x + a₀   is known as a polynomial in variable x.

We use the notations like f(x), h(x), g(x) to denote polynomial in variable x.

Terms and Their Coefficients: If g(x) = an xᶯ + an-1. xᶯ⁻¹ + an-2. xᶯ⁻² +..........+ a₁x + a₀   is a polynomial in variable x, then an xᶯ, an-1. xᶯ⁻¹, an-2. xᶯ⁻², ....., a₁x and a₀ are known as the terms of polynomial g(x) and  an , an-1, an-2, ....., a₁ and a₀ respectively are known as their coefficients.

Here, the coefficient an  of the highest degree term is called the leading coefficient and a₀ is called the constant term.

Example: g(x) = 3x³ + 2x² + 5x + 1

Here, 3x³ , 2x² , 5x , 1 are its terms and 3,2,5,1 are coefficient of x³, x², x and constant term respectively.

Degree of a polynomial: The exponent of the highest degree term in a polynomial is known as its degree. Example,

g(x) = 3x³ + 2x² + 5x + 1 is the polynomial of degree 3.

On the basis of degree of a polynomial, there are different types of polynomials.

Constant Polynomial: A polynomial of degree zero is called a constant polynomial.

Example: g(x) = 8

Linear Polynomial: A polynomial of degree one is called a linear polynomial.

Example: g(x) = 5x  or,  g(x) = 3x+2

Quadratic polynomial: A polynomial of degree two is called quadratic polynomial.

Example: g(x) = 2x² + 5x + 1

Cubic Polynomial: A polynomial of degree three is called a cubic polynomial.

Example: g(x) = 3x³ + 2x² + 5x + 1

Bi-Quadratic Polynomial: A fourth degree polynomial is called a bi-quadratic polynomial.

Example:  3x⁴ + 3x³ + 2x² + 5x + 1


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