Today´s Goal: We describe various methods to factor
algebraic expressions (specifically
polynomials) as a product of simpler ones.
We have used the distributive property to "expand" algebraic
expressions (see Activity 5).
We sometimes need to reverse this process by "factoring" an expression as a product of simpler ones.
Common Factors: The easiest type of factoring occurs when
the terms have a common factor.
Example 1: Factor out the common factor in each of the following expression:
Factoring Trinomials: To factor a trinomial of the form
we need to choose numbers r and s so that r + s = b and r s = c .
Example 2: Factor the following trinomials:
we need to choose numbers p, q, r and s so that p q = a, r s
= c, and q r + p s = b.
That is, p and q are factors of a whereas r and s are factors of c.
Example 3: Factor the following trinomial:
Special Factoring Formulas:
The first three formulas below are simply Special Product Formulas written backward.
If A and B are any real numbers or algebraic expressions, then:
Example 4: Use a Factoring Formula to factor the following expressions:
Factoring by Grouping Terms:
Polynomials with at least four terms can sometimes be factored by grouping terms.
Example 5: Factor the following expressions by grouping terms:
Factoring an Expression Completely:
When we factor an expression, the result can sometimes be factored further. We repeatedly use the methods
outlined in this activity until we have factored our algebraic expression completely.
Example 6: Factor the following expressions completely:
This last problem is trickier! (Hint: add and subtract "4x2" what do you obtain?)
Example 7 (Mowing a Field):
A square field in a certain state park is mo wed around the edges every week. The rest of the field is kept
unmowed to serve as a habitat for the birds and small animals. the field measures b feet by b feet, and the mowed
strip is x feet wide.
(a) Show that the area of the mowed portion is b2-(b-2x)2.
(b) Factor the expression in (a) to show that the area of the mowed portion is also 4x(b-x).