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.

(an even number, please)