Polynomial Concept

Structure of a Monomial

A monomial is a coefficient multiplied by variables whose exponents are nonnegative integers. Compare 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.

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.

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 ($-1$).

  • $5a^{-6}$:

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

A monomial has non-negative integer exponents on every variable. The number multiplying the variables, such as $2$ in $2x^2y$, is called the coefficient.

Definition of Polynomial

A polynomial is a finite sum of monomials. A single monomial is also a polynomial.

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

The definition uses a sum of monomials, while example 1 is written with subtraction $4x^3y - 3x^2$. There is no conflict between those forms.

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.

Exercise

Decide whether each expression is a polynomial in the stated variable. Explain which exponent rule determines your answer. Check the exponent of every variable in each term, including the terms written as a fraction or a root, before you decide.

1. $7x^4-2x+9$ in $x$
2. $\sqrt{x}+1$ in $x$
3. $\frac{3}{x^2}-5$ in $x$
4. $5y^2z^3-2yz+4$ in $y$ and $z$

Worked Solution

Each step below names the deciding condition, so the exponent rule that settles the answer appears next to every expression.

1. Polynomial. The exponents of $x$ are $4$, $1$, and $0$.
2. Not a polynomial. Since $\sqrt{x}=x^{1/2}$, the exponent $1/2$ is not a non-negative integer.
3. Not a polynomial. Since $\frac{3}{x^2}=3x^{-2}$, the exponent $-2$ is negative.
4. Polynomial. Every exponent of $y$ and $z$ is a non-negative integer. A term may omit a variable because that variable then has exponent $0$.