Factoring Simple Expressions

DEFINITION: to factor an expression

To factor an expression means to take the expression and rename it as a product.

That is, to factor an expression means to write the expression as a product.

EXAMPLES:

Question: Factor:

\(\, ab + ac\)

Solution:

\(ab + ac = a(b + c)\)
The expression

\(\,ab + ac\,\) is a sum, since the last operation is addition.
The expression

\(\,a(b + c)\,\) is a product, since the last operation is multiplication.
The process of factoring took us from the sum

\(\,ab + ac\,\) to the product

\(\,a(b + c)\,\).

Notice that \(\,\,ab + ac = a(b + c)\,\,\) is just the distributive law, backwards!

In going from the name \(\,ab + ac\,\) to the name \(\, a(b + c) \,\),
the common factor (\(\,a\,\)) is first identified, and written down.
Next, an opening parenthesis ‘ ( ’ is inserted.
Then, the remaining parts of each term are written down.
Finally, the closing parenthesis ‘ ) ’ is inserted.

Question: Write in factored form:

\(\,3x - 3t\,\)

Solution:

\(3(x - t)\)

Question: Write in factored form:

\(\,2xy - 2yz\)

Solution:

\( 2y(x - z)\)

Question: Write in factored form:

\(\,5x^2 - x^2y^2\)

Solution:

\( x^2(5 - y^2) \)
Note: In the exercises below, exponents are typed in using the ‘^’ key.
For example,

\(\, x^2(5 - y^2) \,\) is typed in as x^2(5 - y^2) .

Question: Write in factored form:

\(\,x(2x + 1) - 3(2x + 1)\)

Solution:

\( (2x + 1)(x - 3) \)
Note: The product

\(\,(2x+1)(x-3)\,\) can also be written as

\(\,(x-3)(2x+1)\,\).
There is no convention here about which name is ‘best’.
The exercise below recognizes both answers.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Listing All the Factors of a Whole Number

(an even number, please)