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.