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It is possible to simplify an algebraic expression by multiplying its terms. Multiplying two polynomials together is the same as multiplying each of the first polynomial’s terms by each of the second.
There are three steps to perform when multiplying algebraic expressions:
Reduce the expression, if necessary, by adding or subtracting like terms (prior to multiplying).
Perform the multiplication.
Reduce the resulting expression, if necessary, by adding or subtracting like terms.
When multiplying algebraic expressions, two important rules must be followed, which are based on the commutative property of multiplication.
A. Multiply the numbers together and the variables together. ||3x \times 4y = 3 \times 4 \times x\times y = 12xy|| B. When multiplying two identical variables together, add their exponents. ||x^2y^3\times x^3y^7 = x^2 \times x^3 \times y^3\times y^7 = x^{(2+3)}\times y^{(3+7)} = x^5y^{10}||
All terms, whether they are like terms or not, can be multiplied together. However, only like terms can be added or subtracted together.
Rarely do equations consist only of multiplications, but when this does occur, make sure to respect the order of operations when reducing the algebraic expression.
To multiply algebraic expressions, it is essential to master the properties and laws of exponents. The distributive property must be applied as well. When multiplying algebraic expressions, several situations can occur:
When multiplying a monomial by a monomial, first multiply the coefficients together and then add together the exponents of identical variables.
When multiplying a constant term by a monomial, multiply the coefficient of the monomial by the constant term.
Let the constant term be |-3| and the monomial be |4xy^2|.
Perform the multiplication |-3\times 4xy^2|.
Multiply the constant term by the coefficient of the monomial:
||-3 \times 4 = -12||
Write the final answer by adding the variables that were temporarily set aside:
||-3\times 4xy^2 = -12xy^2||
When multiplying two monomials together, multiply the coefficients of the two monomials and add together the exponents of identical variables.
Consider the following two monomials: |-3x^3y^4| and |4xy^2|.
Carry out the multiplication of |-3x^{3}y^4\times 4xy^2|.
Multiply the coefficients together:
||-3\times 4 = -12||
Add the exponents of identical variables:
||x^{(3+1)}\quad \text{and}\quad y^{(4+2)}||
Write the final answer:
||-3x^{3}y^4\times 4xy^{2} = -12x^{4}y^{6}||
Here are the steps in detail:
||\begin{align} -3x^{3}y^4 \times 4xy^2 &= (-3\times {4}) {(x^{3}\times {x})}{(y^{4}\times {y^{2}})}\\
&=(-12){(x^{3+1})}{(y^{4+2})}\\
&=(-12){(x^{4})}{(y^{6})}\\
&=-12x^{4}y^{6}\end{align}||