Polynomials

Question 1

What is a polynomial?

Answer 1

A polynomial is an algebraic expression in which the only arithmetic is addition, subtraction, multiplication and non-negative integer exponents of the variables.
Types:
1. polynomial with one variable
2. polynomial with two variable
3. polynomial with more than two variable.

Polynomials are functions from Real number (domain) to Rela numbers (co-domain)

Question 2

How are polynomials represented?

Answer 2

Summation over m=0 to n , am(x^m)

Question 3

what is the degree of the polynomial?

Answer 3

Exponent on the variable in a term is called the degree of that variable in that term.
The degree of that term is the sum of the degrees of the variables in that term.
The degree of the polynomial is the largest degree of any one of the terms with non-zero coefficients

Question 4

What are the classifications of polynomials based on degree of the polynomial?

Answer 4

degree = 0 (Constant Polynomial)
degree =1 (Linear polynomial)
degree =2 (Quadratic polynomial)
degree=3 (Cubic polynomial)
degree = 4 (Quartic Polynomial)

Question 5

Algebra of polynomials?

Answer 5

Addition: Add same power terms, keep different power terms as is. similarly for subtraction.
Multiplication of polynomials: multiply each power term with each and every power term of the second polynomial and then add the power terms
Division of polynomials: regular division, except the numerator will be the next term required for cancelling the first remaining power term.

Question 6

what is the division algorithm for polynomials?

Answer 6

Step1: Arrange the terms in descending order of the degree and add the missing exponents with 0 as coefficient.
Step2: Divide the first term of the dividend by the first term of the divisor and get the monomial
Step3: Multiply the monomial with divisor and subtract the result from the dividend
Step 4: Check if the resultant polynomial has degree less than divisor. If true, write the remainder else Go to Step 2.

Question 7

What are the characteristics of roots of polynomials?

Answer 7

Theorem: k is a root of the polynomial p(x) if and only if (x-k) is a factor of p(x).

Simply , find the number which will make P(k) =0 to find the roots of the polynomial.