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:
[beautiful math coming... please be patient]$\, ab + ac$
Solution:
[beautiful math coming... please be patient]$ab + ac = a(b + c)$
The expression
[beautiful math coming... please be patient]$\,ab + ac\,$ is a sum, since the last operation is addition.
The expression
[beautiful math coming... please be patient]$\,a(b + c)\,$ is a product, since the last operation is multiplication.
The process of factoring took us from the sum
[beautiful math coming... please be patient]$\,ab + ac\,$
to the product
[beautiful math coming... please be patient]$\,a(b + c)\,$.
Notice that
[beautiful math coming... please be patient]$\,\,ab + ac = a(b + c)\,\,$
is just the distributive law, backwards!
In going from the name
[beautiful math coming... please be patient]$\,ab + ac\,$
to the name
[beautiful math coming... please be patient]$\, a(b + c) \,$,
the common factor
([beautiful math coming... please be patient]$\,a\,$) 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: [beautiful math coming... please be patient]$\,3x - 3t\,$
Solution:
[beautiful math coming... please be patient]$3(x - t)$
Question:
Write in factored form: [beautiful math coming... please be patient]$\,2xy - 2yz$
Solution:
[beautiful math coming... please be patient]$ 2y(x - z)$
Question:
Write in factored form: [beautiful math coming... please be patient]$\,5x^2 - x^2y^2$
Solution:
[beautiful math coming... please be patient]$ x^2(5 - y^2) $
Note: In the exercises below, exponents are typed in using the ‘^’ key.
For example, [beautiful math coming... please be patient]$\, x^2(5 - y^2) \,$ is typed in as x^2(5 - y^2) .
Question:
Write in factored form: [beautiful math coming... please be patient]$\,x(2x + 1) - 3(2x + 1)$
Solution:
[beautiful math coming... please be patient]$ (2x + 1)(x - 3) $
Note: The product
[beautiful math coming... please be patient]$\,(2x+1)(x-3)\,$ can also be written as
[beautiful math coming... please be patient]$\,(x-3)(2x+1)\,$.
There is no convention here about which name is ‘best’.
The exercise below recognizes both answers.