3.4 - Multiplication of Expressions

Before studying this topic you may wish to review the multiplication of numbers and the properties of exponents.

Suppose that A and B are any two expressions. Multiplying A and B means setting up the product as (A) · (B), and then actually carrying out the multiplication.

The reason that we put brackets around A and B is that they are expressions, not just numbers, and the multiplication is supposed to apply to whatever A and B may contain. (We want the multiplication to be at the end in the order of operations.)

How the multiplication is actually carried out depends on whether A and B are monomials or multinomials:



Case I: multiplying two monomials

In this case we are multiplying monomials, and you may recall that monomials consist of factors that are already multiplied together. The associative law says that it doesn't matter in what order we carry out all the multiplications. Thus the brackets around the monomials can simply be dropped. (However we must pay attention to the rules for signs and move any minus sign to the front.)

Once the brackets have been removed then we simplify as much as possible. The commutative law is used to move the coefficients to the front and to rearrange the literals (letters) in alphabetical order. Then coefficients are multiplied, and like factors are combined using the properties of exponents.


Example: Multiply and simplify  

Example: Multiply and simplify  

Example: Multiply and simplify  

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Case II: Multiplying a monomial by a multinomial

In this case the distributive law is used to remove the brackets. The monomial is distributed over each term of the multinomial. Then each resulting term is simplified as in Case I above.

Note that the Algebra Coach does not distribute a monomial over a multinomial when you click the Simplify button. You must use the Distribute button to do that. The reason is that the factored form of an expression is usually considered to be simpler than the distributed form.

Example: Multiply and simplify  
Use the distributive law to remove the brackets.
Simplify each term using the properties of exponents.


Example: Multiply and simplify  
Distributing a negative monomial causes the signs in the multinomial (+ − −) to reverse (to become − + +).
Simplify each term using the properties of exponents.

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Case III: Multiplying two multinomials

In this case the distributive law is used to remove the brackets. Each term of the first multinomial is distributed over each term of the second multinomial. Then each resulting term is simplified as in Case I above. Finally, any like terms are combined.

Note that, as in case II, the Algebra Coach does not distribute a multinomial over a multinomial when you click the Simplify button. You must click the Distribute button (several times) to do that. The reason is that the factored form of an expression is usually considered to be simpler than the distributed form.

Example: Multiply and simplify  
It is a good idea to show this “baby step” distribution where the second multinomial is kept together.
Distribute twice more and line up the like terms.
Combine the like terms.


Example: Multiply and simplify  
Distribute using the foil method.

After canceling like terms, the result is a so-called difference of squares.

Example: Multiply and simplify  
Write the square as the multinomial multiplied by itself.

Distribute using the foil method.

Simplify. Compare this with the previous example. Here the like terms add instead of cancel.

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