Monomial and Binomial Expressions

As emphasized previously, the distributive property proves important when carrying out multiplication operations. When you multiply a polynomial by an expression consisting of only one term, you apply the single term to each of the terms in the polynomial expression. Here is an example:

When you work with binomial expressions, the same practice applies. You must apply both of the terms in the first expression to both of the terms in the second expression. A common form of binomial multiplication problem involves two binomials that add terms:

As Figure 8.2 illustrates, you can visualize a binomial multiplication problem if you use a series of blocks to represent the values the multiplication involves.

You also commonly encounter binomial problems in which the first expression adds terms, while the second expression subtracts one term from another. You employ the same procedure as before. Using distribution, you multiply each term in the second expression by each term in the first:

(5m + 4)(m − 3)

5m (m) − 5m (3)+ 4(m)− 4(3)

5m² − 15m + 4m − 12

5m² − 11m − 12

In this instance, the two middle terms are of opposite signs, and whether the final middle term is positive depends on the values of the terms combined.

A third form of binomial that occurs commonly involves two expressions that are characterized by subtraction. Here is an example:

(3x − 5)(4x − 3)

3x(4x) − 3x(3)− 5(4x)−(5)(−3)

12x² − 9x − 20x + 15

12x² − 29x + 15

In such expressions, the multiplication of a negative by a negative value generates a positive value. The middle term, likewise, is negative, because when you create the values for the middle term, you combine two negative terms.