Chapter 17: Factoring

Factoring is the process of resolving a composite expression into its original factors. In previous chapters, we learned how to multiply expressions together; now, we learn how to work backward. Mastering factoring is essential for simplifying complex fractions and solving quadratic equations.

Wentworth identifies several distinct cases of factoring in this chapter:

17.1 Common Monomial Factors

The simplest form of factoring is identifying a factor that is common to every term in the expression. This is essentially the Distributive Property in reverse.

Example:
Factor: 3x²y + 6xy²

Identify the greatest common factor:
Both terms are divisible by 3, x, and y. Therefore, the common factor is 3xy.

Divide each term by 3xy:
3x²y / 3xy = x
6xy² / 3xy = 2y

Result: 3xy(x + 2y)

17.2 Factoring by Grouping

Sometimes there is no factor common to every term, but we can group terms together that do share common factors.

Example:
Factor: ax + ay + bx + by

Group the first two and last two terms:
(ax + ay) + (bx + by)

Factor each group:
a(x + y) + b(x + y)

Now, factor out the common binomial (x + y):
Result: (a + b)(x + y)

17.3 The Difference of Two Squares

One of the most recognizable patterns in algebra is the difference of two perfect squares. It always factors into the sum and difference of the square roots.

a² − b² = (a + b)(a − b)

Example:
Factor: 25x² − 16

Identify the square roots:
√25x² = 5x
√16 = 4

Result: (5x + 4)(5x − 4)

17.4 Trinomials (x² + ax + b)

To factor a trinomial like x² + 5x + 6, we look for two numbers that multiply to give the last term (6) and add to give the middle coefficient (5).

Example:
Factor: x² + 7x + 10

Find two numbers that multiply to 10 and add to 7:
Factors of 10 are (1, 10) and (2, 5).
2 + 5 = 7. These are our numbers.

Result: (x + 2)(x + 5)

Chapter 17 Exercises