Polynomial and its Types
The highest exponent of the variable x in a polynomial p(x) is called the degree of the polynomial p(x). The standard form of a polynomial in variable x can be represented as p(x) = a0 + a1x + a2x2 + . . . + anxn , where n is a positive integer,
a0, a1 a2 . . . an are constants and an ≠ 0.
A polynomial of degree one is called a linear polynomial. A polynomial of degree two is called a quadratic polynomial. A polynomial of degree three is called a cubic polynomial. A polynomial having only one term is called monomial. A polynomials having only two terms are called binomial. A polynomial having only three terms is called trinomial.
Value of a Polynomial
The value of a polynomial p(x) at x = a is obtained by substituting x = a in the given polynomial and is denoted by p(a).
Graph of a polynomial
The graph of a polynomial p(x) is the collection of all points (x, y), where y = p(x). For the linear polynomial ax + b (a 0), the graph of y= ax+b is a straight line and for a quadratic polynomial ax2 + bx + c (a 0), the graph of y= ax2 + bx + c is a parabola.
Zeroes of a Polynomial
A real number k is a zero of a polynomial p(x) if p(k) = 0.
Geometrical Meaning of zeroes of a Polynomial
A linear polynomial has exactly one zero. The graph of y= ax + b, a 0 intersects the x-axis at exactly one point so, the x-coordinate of this point is a zero of linear polynomial ax + b. A quadratic polynomial can have at most two zeroes.
If the graph of y = ax2 + bx + c cuts the x-axis at two distinct points, the
x-coordinates of these points are the two zeroes of the quadratic polynomial
ax2 + bx + c.
If the graph of y= ax2 + bx + c cuts the x-axis at exactly one point, the x-coordinate of this point is the only zero of the quadratic polynomial ax2 + bx + c.
If the graph does not cut the x-axis at all, the quadratic polynomial ax2 + bx + c has no zero.
Relation between Zeroes and Coefficients of a Polynomial
Zero of a linear polynomial ax + b is .
A cubic polynomial has at most three zeroes. In general, a polynomial of degree n can have at most n zeroes
Division Algorithm for Polynomials
If p(x) and g(x) are any two polynomials with g(x) 0, then there are unique polynomials q(x) and r(x) such that ,
where r (x) = 0 or degree r (x) < degree g (x).