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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