Polynomial Concept

Getting to Know Monomials

Before defining polynomials, start with their building blocks: monomials. Consider the following algebraic expressions:

From the expressions above, we can group them into two:

1. Group $1$ (Monomials): $2x^2y$, $-8$, $1.24k^4$
2. Group $2$ (Not Monomials): $\sqrt[3]{p}$, $\frac{2}{m}$, $5a^{-6}$

The algebraic expressions in Group $1$ are what we call monomials.

What is a Monomial?

A monomial is a number, a variable raised to a non-negative integer power ($0, 1, 2, 3, \dots$), or the product of a number and one or more variables raised to non-negative integer powers.

Let's break down why Group $1$ consists of monomials and Group $2$ does not:

• Group $1$ (Monomials):

  • $2x^2y$:

    The product of a number ($2$) and variables ($x$, $y$) with non-negative integer powers ($2$ and $1$).

  • $-8$:

    A constant (just a number). Or this is the same as $-8x^0$.

  • $1.24k^4$:

    The product of a number ($1.24$) and a variable ($k$) with a non-negative integer power ($4$).

• Group $2$ (Not Monomials):

  • $\sqrt[3]{p} = p^{1/3}$:

    The power of the variable $p$ is not a non-negative integer ($1/3$).

  • $\frac{2}{m} = 2m^{-1}$:

    The power of the variable $m$ is not a non-negative integer ($-1$).

  • $5a^{-6} = 5a^{-6}$:

    The power of the variable $a$ is not a non-negative integer ($-6$).

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 $2$ in $2x^2y$) is called the coefficient.

Definition of Polynomial

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:

1. $4x^3y - 3x^2$
   • The term $4x^3y$ is a monomial.
   • The term $-3x^2$ is a monomial.
   • Conclusion: Polynomial (subtraction of two monomials).
2. $x + 2\sqrt{x}$
   • The term $x$ is a monomial.
   • The term $2\sqrt{x} = 2x^{1/2}$ is not a monomial (exponent is not a non-negative integer).
   • Conclusion: Not a Polynomial.
3. $2x^3 - 5x^{-2} + 1$
   • The term $2x^3$ is a monomial.
   • The term $-5x^{-2}$ is not a monomial (exponent is not a non-negative integer).
   • The term $1$ is a monomial (constant).
   • Conclusion: Not a Polynomial.

Addition and Subtraction in Polynomials

You might ask, "The definition of a polynomial involves the sum of monomials, but example $1$ has subtraction ($4x^3y - 3x^2$). How does that work?"

Recall that subtraction can be viewed as adding the negative. So, $4x^3y - 3x^2$ is the same as $4x^3y + (-3x^2)$.

Since both $4x^3y$ and $-3x^2$ 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.