Getting to Know Monomials
Before defining polynomials, start with their building blocks: monomials. Consider the following algebraic expressions:
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Before defining polynomials, start with their building blocks: monomials. Consider the following algebraic expressions:
From the expressions above, we can group them into two:
The algebraic expressions in Group are what we call monomials.
A monomial is a number, a variable raised to a non-negative integer power (), or the product of a number and one or more variables raised to non-negative integer powers.
Let's break down why Group consists of monomials and Group does not:
Group (Monomials):
:
The product of a number () and variables (, ) with non-negative integer powers ( and ).
:
A constant (just a number). Or this is the same as .
:
The product of a number () and a variable () with a non-negative integer power ().
Group (Not Monomials):
:
So, the key characteristic of a monomial is that the exponents of the variables must be non-negative integers. The number multiplying the variable (like in ) is called the coefficient.
After understanding monomials, we can now define a polynomial.
A polynomial is an algebraic expression that is a monomial or the sum (and subtraction) of two or more monomials.
Consider the following examples:
Let's identify which are polynomials and which are not:
You might ask, "The definition of a polynomial involves the sum of monomials, but example has subtraction (). How does that work?"
Recall that subtraction can be viewed as adding the negative. So, is the same as .
Since both and are monomials, their sum is still a polynomial. This is why subtraction between monomials also results in a polynomial.
So, an algebraic expression is called a polynomial if all its terms are monomials, meaning every variable has a non-negative integer exponent.
The power of the variable is not a non-negative integer ().
:
The power of the variable is not a non-negative integer ().
:
The power of the variable is not a non-negative integer ().