A monomial is a single algebraic expression consisting of a numerical coefficient, variables, and exponents. Monomials are the building blocks of polynomials and are essential in understanding operations like adding, subtracting, multiplying, and dividing polynomials.
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Monomials can be added or subtracted by combining like terms, which have the same variable(s) and exponents.
The multiplication properties of exponents, such as $a^m \cdot a^n = a^{m+n}$, are used when multiplying monomials.
Multiplying monomials involves multiplying the coefficients and adding the exponents of the corresponding variables.
Dividing monomials involves dividing the coefficients and subtracting the exponents of the corresponding variables.
Dividing polynomials is done by first expressing the polynomial as a sum of monomials and then dividing each monomial term.
Review Questions
How can monomials be added or subtracted?
Monomials can be added or subtracted by combining like terms, which have the same variable(s) and exponents. To add or subtract monomials, you simply add or subtract the coefficients of the like terms. For example, $3x^2 + 5x^2 = 8x^2$ and $4y^3 - 2y^3 = 2y^3$.
Explain the multiplication properties of exponents and how they are used when multiplying monomials.
The multiplication properties of exponents state that $a^m \cdot a^n = a^{m+n}$ and $(a^m)^n = a^{mn}$. These properties are crucial when multiplying monomials because they allow you to simplify the exponents. For instance, to multiply $2x^3 \cdot 4x^2$, you would multiply the coefficients (2 and 4) to get 8, and then add the exponents of the variable x (3 and 2) to get $8x^5$.
Describe the process of dividing polynomials and how monomials are involved in this operation.
To divide a polynomial by another polynomial, you first need to express the polynomial as a sum of monomials. Then, you can divide each monomial term in the numerator by the monomial terms in the denominator. This involves dividing the coefficients and subtracting the exponents of the corresponding variables. For example, to divide $12x^4y^2 - 6x^2y^3$ by $3x^2y$, you would first express the polynomial as $4x^2y - 2xy^2$, and then divide each monomial term by $3x^2y$, resulting in $4 - 2/x$.