INTRODUCTION:

INTRODUCTION:

 The value of the polynomial p(x) at x = a is p(a).

 Zeroes of the polynomial p(x) can be found by equating p(x) to zero and solving the equation.

 If for p(x) = ax² + bx + c = 0,   a ≠ 0;    α and β are the zeroes, then

 If α and β are the zeroes; then quadratic polynomial will be given by k [x² – Sx + P]
where  S = α +β   and  P = αβ     k (≠0) is real.

 The degree of a polynomial p(x) in x is the highest power of x in p(x)
 (i) Polynomial with degree 1, of the form ax + b;   a ≠ 0 is called linear polynomial.
(ii) Polynomial with degree 2, of the form ax² + bx + c;   a ≠ 0 is called quadratic polynomial.
(iii) Polynomial with degree 3, of the form ax³ + bx² + cx + d ;   a ≠ 0 is called cubic polynomial.

 For any polynomial y = f(x), the number of points on which the graph of y = f(x) intersects at x-axis is called the number of the zeroes of the polynomial and the x-coordinates of these points are called the zeroes of the polynomial y = f(x).

 Polynomial with degree ‘n’ has maximum ‘n’ number of zeroes.
 A constant polynomial has no zeroes.
Geometrical representation of a linear polynomial is always a straight line.

 Geometrical representation of a quadratic polynomial is the graph of the shape either open upwards like ‘∪’ or open downwards like ‘∩’ according to a > 0 or a < 0. These curves are called Parabolas.

 If α and β are the zeroes of the quadratic polynomial  p(x) = ax² + bx + c    where  a≠0 then

 A quadratic polynomial p(x) with zeroes α and β is given by
p(x) = k [x² – (α + β)x + αβ]
where k (≠0) is real.

Division Algorithm for Polynomials:
If p(x) and g(x) are any two polynomials where g(x) ≠ 0.
Then on dividing p(x) by g(x), we can find other two polynomials q(x) and r(x) such that
p(x) = g(x) x q(x) + r(x);
where deg. of r(x) < deg. of g(x)
or Dividend = (Divisor x Quotient) + Remainder

Note:

• If r(x) = 0, then g(x) will be a factor of p(x) otherwise not.
• If any real number ‘a’ is a zero of the polynomial p(x), then (x – a) will be a factor of p(x).