The Language of Factoring
Definitions: Product, Factors, Sum, Terms
A product is an expression where the last operation is multiplication. In a product, the things being multiplied are called the factors.
A sum is an expression where the last operation
is addition. In a sum, the things being added are called the terms.
1. For example, consider the expression $$a(b+c)$$.
If numbers are chosen for $$a$$, $$b$$, and $$c$$, here is the order we would use to do the computations:
- Add $$b$$ and $$c$$
- Pre-multiply this sum by $$a$$
Notice that the last operation done is multiplication. Thus, the expression $$a(b+c)$$ is a product.
The factors are $$a$$ and $$(b+c)$$.
2. As a second example, consider the expression $$ab+c$$.
Given numbers $$a$$, $$b$$, and $$c$$, here is the order we would use to do the computations:
- Multiply $$a$$ and $$b$$
- Add this result to $$c$$
Notice that the last operation we do is addition. Thus, the expression $$ab+c$$ is a sum.
The terms are $$ab$$ and $$c$$.
Examples
1. The expression $$3xy$$ is a product.
The factors are $$3$$, $$x$$, $$y$$
Note: The factors must be listed in order from left to right, and must be separated by commas.
2. The expression $$-4x(x+2)$$ is a product.
The factors are $$-4$$, $$x$$, $$x+2$$
Note: Do not use parentheses when listing factors. In other words, do not put the $$x+2$$ inside parentheses.
3. The expression $$5x-y+1$$ is a sum.
The terms are $$5x$$, $$-y$$, $$1$$
Note: The terms must be listed in order from left to right, and must be separated by commas.
Remember that a term includes its sign.
4. The expression $$x^{2}+2y^{3}-7$$ is a sum.
The terms are $$x^{2}$$, $$2y^{3}$$, $$-7$$
Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/prod_sum.htm
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