Polynomials

A polynomial is an expression consists of constants, variables and exponents. It’s mathematical form is-

a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + a₂x² + a₁x + a₀ = 0

where the (a_i)’s are constant

Degree of Polynomials

Let P(y) is a polynomial in y, then the highest power of y in the P(y) will be the degree of polynomial P(y).

Types of Polynomial according to their Degrees

Type of polynomial	Degree	Form
Constant	0	P(x) = a
Linear	1	P(x) = ax + b
Quadratic	2	P(x) = ax2 + ax + b
Cubic	3	P(x) = ax3 + ax2 + ax + b
Bi-quadratic	4	P(x) = ax4 + ax3 + ax2 + ax + b

Value of Polynomial

Let p(y) is a polynomial in y and α could be any real number, then the value calculated after putting the value y = α in p(y) is the final value of p(y) at y = α. This shows that p(y) at y = α is represented by p (α).

Zero of a Polynomial

If the value of p(y) at y = k is 0, that is p (k) = 0 then y = k will be the zero of that polynomial p(y).

Geometrical meaning of the Zeros of a Polynomial

Zeroes of the polynomials are the x coordinates of the point where the graph of that polynomial intersects the x-axis.

Graph of a Linear Polynomial

Graph of a linear polynomial is a straight line which intersects the x-axis at one point only, so a linear polynomial has 1 degree.

Graph of Quadratic Polynomial

Case 1: When the graph cuts the x-axis at two points than these two points are the two zeroes of quadratic polynomials.

Case 2: When the graph cuts the x-axis at only one point then that particular point is the zero of that quadratic polynomial and the equation is in the form of a perfect square

Case 3: When the graph does not intersect the x-axis at any point i.e. the graph is either completely above the x-axis or below the x-axis then that quadratic polynomial has no zero as it is not intersecting the x-axis at any point.

Hence the quadratic polynomial can have either two zeroes, one zero or no zero. Or you can say that it can have maximum two zero only.

Relationship between Zeros and Coefficients of a Polynomial

Division Algorithm for Polynomial

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that

P(x) = g(x) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x).