Polynomials

Polynomials 

-November 15,2024

On the first day of micro teaching , the topic I choose to teach is Polynomials 

A polynomial is an arithmetic expression consisting of variables and constants that involves four fundamental arithmetic operations and non-negative integer exponents of variables. 

Polynomial in One Variable 

An algebraic expression of the form 

P(X)=a_nxⁿ+a_n-1x^n-1+…..+a₂x²+a₁x+a₀ is called Polynomial in one variable x of degree ‘n’ where a₀, a₁,a₂,….,a_n  are constants (a_n≠0) and n is a whole number. 

In general polynomials are denoted by f(x), g(x), p(t), q(z) and r(x) and so on.

The coefficient of variables in the algebraic expression may have any real numbers, where as the powers of variables in polynomial must have only non-negative integral powers that is, only whole numbers. Recall that  a₀= 1 for all a.

Standard Form of a Polynomial

We can write a polynomial p(x) in the decreasing or increasing order of the powers of x. This way of writing the polynomial is called the standard form of a polynomial.

Example 1:

Polynomial:7m²+5m⁴-3m+8

Standard form:5m⁴+7m²-3m+8

Example 2:

Polynomial:12p²+8p⁵-10p⁴-7

Standard form:8p⁵-10p⁴+12p²-7

Degree of the Polynomial

In a polynomial of one variable, the highest power of the variable is called the degree of the polynomial.

In case of a polynomial of more than one variable, the sum of the powers of the variables in each term is considered and the highest sum so obtained is called the degree of the polynomial.

This is intended as the most significant power of the polynomial. Obviously when we write  x²+5x  the value of x² becomes much larger than 5x for large values of x. So we could think of x²+5x  being almost the same as x² for large values of x. So the higher the power, the more it dominates. That is why we use the highest power as important information about the polynomial and give it a name.

Example:

Find the degree of each term for the following polynomial and also find the degree of the polynomial

6ab⁸+5a²b³c²-7ab+4b²c+2.

Solution:

Given polynomial is 

6ab⁸+5a²b³c²-7ab+4b²c+2

Degree of each of the terms is given below.

6ab⁸ has degree (1+8) = 9

5a²b³c²has degree (2+3+2) = 7

7ab has degree (1+1) = 2

4b²c has degree (2+1) = 3

The constant term 2 is always regarded as having degree Zero.

The degree of the polynomial 6ab⁸+5a²b³c²-7ab+4b²c+2 

                                                = the largest exponent in the polynomial

                                                = 9

A very Special Polynomial

We have said that coefficients can be any real numbers. What if the coefficient is zero? Well that term becomes zero, so we won’t write it. What if all the coefficients are zero? We acknowledge that it exists and give it a name. 

It is the polynomial having all its coefficients to be zero.

g(t)=0t⁴+0t²-0t,

h(p)= 0p²-0p+0

From the above example we see that we cannot talk of the degree of the zero polynomial, since the above two have different degrees but both are zero polynomial. So we say that the degree of the zero polynomial is not defined.

Evaluation:

1.what is polynomials?

Ans: A polynomial is an arithmetic expression consisting of variables and constants that involves four fundamental arithmetic operations and non-negative integer exponents of variables. 

2.what is the degree of the polynomial 4p³-3p+0 ?

Ans: Degree of the polynomial is 3

3.what is the degree of 14ab?

Ans: 14ab has degree (1+1)=2.

4.Give the standard form for the polynomial 12t⁴+7t⁵+t-t².            

Ans:7t⁵+12t⁴-t²+t              

5.what is the degree of the zero

polynomial?

Ans: The degree of the zero polynomial is not defined.