Polynomials

1. What is a polynomial?
   
   A polynomial p(x) in one variable x is an algebraic expression in x of the form
   
2. The highest exponent of the variable in a polynomial determines the degree of the polynomial.
   
   
   
3. Types of polynomials
   

   a. A polynomial of degree zero is called a constant polynomial. 
   
   
   b. A polynomial of degree one is called a linear polynomial. It is of the form ax + b.

       Examples: x – 2, 4y + 89, 3x – z.
   
   c. A polynomial of degree two is called a quadratic polynomial. It is of the form ax² + bx + c, where a, b, c are real numbers and 

   Examples: x²- 2x + 5, x² – 3x etc.
   
   d. A polynomial of degree 3 is called a cubic polynomial and has the general form ax³ + bx² + cx +d.
   
   
   

4. Value of the polynomial
   
   If p(x) is a polynomial in x, and k is a real number then the value obtained after replacing x by k in p(x) is called the value of p(x) at x = k which is denoted by p(k).
   
   
   
5. Zero of a polynomial
   
   a. A real number k is said to be the zero of the polynomial p(x), if p(k) = 0.
   b. Zeroes of the polynomial can be obtained by solving the equation p(x) = 0.
   c. It is possible that a polynomial may not have a real zero at all.
   d. For any linear polynomial ax + b, the zero is given by the expression (–b/a) = –(constant term)/(Coefficient of x).
   
   
   
6. Number of zeroes of a polynomial
   
   a. The number of real zeros of the polynomial is the number of times its graph touches or intersects x-axis.
   b. The graph of a polynomial p(x) of degree n intersects or touches the x-axis at at most n points.
   c. A polynomial of degree n has at most n distinct real zeroes.
   
   
   
7.  A linear polynomial has at most one real zero.
   
   
8.  A quadratic polynomial has at most two real zeroes.
   
   

   Quadratic Polynomial having no zeroes.
   Quadratic Polynomial having one zero.
   Quadratic Polynomial having two zeroes.

   
   
    
9. A cubic polynomial has at most three real zeroes.
   
   

   Cubic Polynomial having no zeroes.
   Cubic Polynomial having one zero.
   Cubic Polynomial having one zeroes.
   Cubic Polynomial having three zeroes.

   
   
   
10.  Relationship between zeroes and coefficients of a polynomial:
    
    i.    For a linear polynomial ax + b, a ¹ 0, the zero is. It can be observed that:
    
    ii.    For a quadratic polynomial ax² + bx + c,
    
    
    iii.    For a cubic polynomial ax³ + bx² + cx + d = 0, 
    
    
    
    
11. The quadratic polynomial whose sum of the zeroes 
    
    
    
    
12. Process of dividing a polynomial f(x) by another polynomial g(x) is as follows:
    
    Step 1: To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor. Then carry out the division process.
    
    Step 2: To obtain the second term of the quotient, divide the highest degree term of the new dividend by the highest degree term of the divisor. Then again carry out the division process.
    
    Step 3: Continue the process till the degree of the new dividend is less that the degree of the divisor. This will be called the remainder.
    
    
    
13. Division Algorithm for polynomials: If f(x) and g(x) are any two polynomials, where g(x) ≠ 0, then there exists the polynomials q(x) and r(x) such that f(x) = g(x) q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x)
    
    So, q(x) is the quotient and r(x) is the remainder obtained when the polynomial f(x) is divided by the polynomial g(x).
    
    
    
14. Factor of the polynomial
    
    If f(x) = g(x) q(x) + r(x) and r(x) = 0, then polynomial g(x) is a factor of the polynomial f(x).
    
    
    
15. Finding zeroes of a polynomial using division algorithm
    
    Division algorithm can also be used to find the zeroes of a polynomial. For example, if ‘a’ and ‘b’ are two zeroes of a fourth degree polynomial f(x), then other two zeroes can be found out by dividing f(x) by (x-a)(x-b).