A monomial is a single algebraic expression consisting of a numerical coefficient, variables, and non-negative integer exponents. It is the most fundamental building block of polynomial expressions, which are central to the topics of adding, subtracting, multiplying, and dividing polynomials.
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Monomials can have one or more variables, and the exponents on those variables must be non-negative integers.
The degree of a monomial is the sum of the exponents on its variables, for example, $5x^2y^3$ has a degree of 5.
Monomials can be combined by adding their coefficients when the variables and exponents are the same, for example, $2x^3 + 3x^3 = 5x^3$.
Multiplying monomials involves multiplying the coefficients and adding the exponents on corresponding variables.
Dividing one monomial by another involves dividing the coefficients and subtracting the exponents on corresponding variables.
Review Questions
How can monomials be combined when adding or subtracting polynomials?
When adding or subtracting polynomials, monomials with the same variable and exponent can be combined by adding or subtracting their coefficients. For example, $2x^3 + 3x^3$ can be combined to form $5x^3$, and $4y^2 - 2y^2$ can be combined to form $2y^2$. This simplification is possible because monomials with the same variable and exponent represent the same algebraic expression.
Explain the process of multiplying two monomials.
To multiply two monomials, you multiply the coefficients and then add the exponents on the corresponding variables. For example, to multiply $3x^2y^3$ and $2x^4y$, you would multiply the coefficients (3 and 2) to get 6, and then add the exponents on the $x$ variable (2 and 4) to get $x^6$, and add the exponents on the $y$ variable (3 and 1) to get $y^4$. The final result is $6x^6y^4$. This process allows you to multiply polynomial expressions by first multiplying the individual monomials.
How can monomials be used to divide polynomials?
When dividing one polynomial by another, you can use the properties of monomials to perform the division. First, identify the monomial that you need to divide the polynomial by. Then, divide the coefficients and subtract the exponents on the corresponding variables. For example, to divide $12x^4y^3$ by $3x^2y$, you would divide the coefficients (12 by 3) to get 4, and then subtract the exponents on the $x$ variable (4 - 2 = 2) and the $y$ variable (3 - 1 = 2), resulting in the quotient $4x^2y^2$. This process allows you to systematically divide one polynomial expression by another using the properties of monomials.