12 Questions
How do you multiply monomials with multiple variables?
By using the distributive property and the rules of exponents to find the exponent for each variable.
If you have the monomials $3 x^{2} y$ and $4 x^{2} z$, what is the result of their multiplication?
12 x^{4} y z
When multiplying a monomial by a binomial or trinomial, what should you do?
Distribute the monomial to each term of the polynomial.
How should you approach multiplying polynomials with multiple variables?
Use the same principles as for monomials, applying the distributive property and rules of exponents.
What steps can you follow to multiply a monomial by a trinomial?
Distribute the monomial to each term of the trinomial.
In the expression $5 x^{2} y^{3}(2 x^{3} + 3 x^{2} y - 4 x y^{2} + 2)$, what is the result of the multiplication?
10 x^{5} y^{3} + 15 x^{4} y^{4} - 20 x^{3} y^{5} + 10 x^{2} y^{3}
How do you multiply two monomials with multiple variables?
To multiply two monomials with multiple variables, multiply the coefficients and add the exponents of each variable.
What is the result of multiplying \(5 x^{2} y^{3}\) by \(4 x^{4} y^{2}\)?
The result is \(20 x^{6} y^{5}\).
How can you multiply a monomial by a binomial?
To multiply a monomial by a binomial, use the distributive property by distributing the monomial to each term of the binomial.
What is the result of multiplying \(5 x^{2} y^{2}\) by \(2 x^{2} + 5 x y - 10\)?
The result is \(10 x^{4} y^{2} + 25 x^{3} y^{3} - 50 x^{2} y^{2}\).
How do you multiply monomials with different variables?
When multiplying monomials with different variables, apply the same principles as for monomials with the same variables by multiplying the coefficients and adding the exponents of each variable.
If you multiply \(3 x^{2} y\) by \(4 x^{2} z\), what is the result?
The result is \(12 x^{4} y z\).
Study Notes
Multiplying Monomials with Multiple Variables
Monomials are algebraic expressions that consist of a single term, which can have multiple variables and a higher degree. To multiply monomials with multiple variables, you can use the same techniques as for polynomials with one variable, but you need to be careful to distinguish between the different variables.
Multiplication of Two Monomials
To multiply two monomials, you multiply the coefficients and use the rules of exponents to find the exponent for each variable. For example, consider the monomials ((5 x^{2} y^{3})) and ((4 x^{4} y^{2})). To multiply these monomials, you can use the distributive property:
[ (5 x^{2} y^{3})(4 x^{4} y^{2}) = (5 \cdot 4) x^{2+4} y^{3+2} = 20 x^{6} y^{5} ]
Multiplication of a Monomial by a Binomial
To multiply a monomial by a binomial, you can use the distributive property in the same way as for polynomials with one variable. For example, consider the monomial (5 x^{2} y^{2}) and the binomial (2 x^{2} + 5 x y - 10). You can distribute the monomial to each term of the binomial:
[ 5 x^{2} y^{2}(2 x^{2} + 5 x y - 10) = (\ldots) + (\ldots) + (\ldots) ]
Multiplying Monomials with Different Variables
When multiplying monomials with different variables, you can apply the same principles as for monomials with the same variables. For example, consider the monomials (3 x^{2} y) and (4 x^{2} z). You can multiply the coefficients and use the rules of exponents to find the exponent for each variable:
[ 3 x^{2} y \cdot 4 x^{2} z = 12 x^{2+2} y z = 12 x^{4} y z ]
Multiplying a Monomial by a Trinomial
To multiply a monomial by a trinomial, you can use the same principles as for multiplying a monomial by a binomial. For example, consider the monomial (5 x^{2} y^{3}) and the trinomial (2 x^{3} + 3 x^{2} y - 4 x y^{2} + 2). You can distribute the monomial to each term of the trinomial:
[ 5 x^{2} y^{3}(2 x^{3} + 3 x^{2} y - 4 x y^{2} + 2) = (\ldots) + (\ldots) + (\ldots) + (\ldots) ]
Multiplying Monomials with Multiple Variables
To multiply monomials with multiple variables, you can follow the same principles as for monomials with one variable. You can use the distributive property and the rules of exponents to find the exponent for each variable. For example, consider the monomials (3 x^{2} y) and (4 x^{2} z). You can multiply these monomials as follows:
[ (3 x^{2} y)(4 x^{2} z) = 12 x^{4} y z ]
Multiplying Polynomials with Multiple Variables
When multiplying polynomials with multiple variables, you can use the same principles as for monomials. You can use the distributive property and the rules of exponents to find the exponent for each variable. For example, consider the polynomials (2 x^{2} + 3 x y - 4) and (4 x^{2} + 5 x y + 6). You can distribute the first polynomial to each term of the second polynomial:
[ (2 x^{2} + 3 x y - 4)(4 x^{2} + 5 x y + 6) = \ldots ]
Conclusion
Multiplying monomials with multiple variables follows the same principles as multiplying monomials with one variable. You can use the distributive property and the rules of exponents to find the exponent for each variable. To multiply a monomial by a binomial or trinomial, you can distribute the monomial to each term of the polynomial. When multiplying polynomials with multiple variables, you can use the same principles to find the product of the coefficients and the exponent for each variable.
Learn how to multiply monomials with multiple variables by applying the distributive property and rules of exponents. Practice multiplying monomials by other monomials, binomials, and trinomials while carefully distinguishing between different variables.
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