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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