Notes on Polynomials

 

Class Notes for Polynomial

1.     A polynomial p(x) in one variable  x is algebraic expression s in x in the form of  p(x) = anxn + an-1xn-1 + an-2xn-2 +…….. a2x2 + a1x1 + a0

Where a0, a1, a2, a3,…… an are constants(whole numbers), and an =/= 0.

a0, a1, a2, a3,…… an are  co efficients of x0, x1,….xn  and n  is called degree of  the polynomial p(x). Each of the anxn , an-1xn-1 , an-2xn-2 ,…….. a2x2, a1x1 and  a0 with an =/= 0 is called a term of the polynomial p(x).

Degree of Polynomial : Highest power of variable x in the polynomial is defined as the degree of the polynomial.

Degree

·        A polynomial of degree 1 is called the linear polynomial

·        A polynomial of degree 2 is called the quadratic polynomial

·        A polynomial of degree 3 is called the cubic  polynomial

Terms

·        A polynomial having only one term is called a monomial.

·        A polynomial having  only two terms is called a binomial.

·        A polynomial having three terms is called a trinomial.

2. A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0,  In this case a is also called root of the equation p(x)=0      

3. Zero

• Every linear polynomial in one variable has a unique zero.

• A non-zero constant polynomial has no zero.

• Every real number is a zero of the zero polynomial. 

4. Remainder theorem : If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial (x - a), then the remainder is p(a).

5. Factor theorem : (x - a) is factor of the polynomial p(x), if p(a) = 0. Also if (x-a) is a factor of p(x) than p(a) = 0.