Call Now: (800) 537-1660

May 22nd

May 22nd

Factoring Expressions

In order to understand the following lesson, you should already know:

(i.) a general method for removing monomial factors from an expression

(ii.) two methods for factoring expressions with very special (but quite commonly occurring) forms:

• trinomials (ax 2 + bx + c)
• difference of two perfect squares (u 2 – v 2 )

All we want to point out here is that the overall strategy for factoring any algebraic expression is to apply the methods listed above in order. So,

(i) first remove all monomial factors from the expression

then

(ii) check whatever is left against any of the special patterns that you know

By following this strategy, the more difficult, tedious, and intuitive analysis of the second type in the list above will always be done on the simplest possible expressions.

We’ve already alluded to this strategy in the previous documents on factoring, and shown one or two brief examples. Here is a few more short examples to help you understand the overall strategy better.

Example 1:

Factor completely: 3x 4 y + 6x 3 y – 45x 2 y.

solution:

We see that 3, x 2 , and y, are each monomial factors of all three terms. So, step (i) in the general strategy leads to

3x 4 y + 6x 3 y – 45x 2 y = 3x 2 y(x 2 + 2x – 15)

Now, we need to apply step (ii) of the general strategy to the expression, x 2 + 2x – 15, remaining after we have accounted for the common monomial factors. This is a trinomial in x, which it may be possible to rewrite in the form

x 2 + 2x – 15 = (x + a)(x + b)

if we can find two whole numbers, a and b, such that

ab = -15 and a + b = -2.

You can easily verify that a = 5 and b = -3 works. Thus

x 2 + 2x – 15 = (x + 5)(x - 3)

Thus, for the original expression we now have

3x 4y + 6x 3y – 45x 2y = 3x 2y(x + 5)(x – 3)

which is factored as completely as is possible. We know that no further factorization is possible, because the factors here which are not monomial factors contain no products at all.

Notice that when it came time to check if any of our special pattern formulas might apply, we needed to check them against the quite simple expression, x 2 + 2x – 15, rather than the much more complicated original expression. To be sure, (x + 5) and (x – 3) are factors of the original expression, but its complexity (due to the presence of common monomial factors in each term) would make it very difficult to recognize these binomial factors in that expression.

Free Tutoring
• Get 30 Minutes of FREE Live Tutoring from Tutor.com if you
BUY NOW!

Testimonials
"I ordered the Algebra Buster late one night when my daughter was having problems in her honors algebra class. After we ordered your software she was able to see step by step how to solve the problems. Algebra Buster definitely saved the day."
Tami Garleff

Home    Why Algebra Buster?    Guarantee    Testimonials    Ordering    FAQ    About Us
What's new?    Resources    Animated demo    Algebra lessons    Bibliography of     textbooks

Copyright © 2009, algebra-online.com. All rights reserved.

Related pages: