Interactive Algebra Review
Based on the algebra reviews in Applied Calculus and Finite Mathematics and Applied Calculus
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3. Multiplying and Factoring Algebraic Expressions
Part A: Multiplying Algebraic Expressions
Note If you are happy with multiplying algebraic expressions, and want to study factoring, go to Part B: Factoring Algebraic Expressions.
One of the most important mathematical tools for multiplying algebraic expressions is the distributive law for real numbers, recalled here.
The Distributive Law for Real Numbers
If a, b, and c are any real numbers, then
a(b + c) = ab + ac 

a(b  c) = ab  ac 
(a + b)c = ac + bc 

(a  b)c = ac  bc 
Examples
x(x + 1) = x^{.}x + x^{.}1 = x^{2} + x 
(y  2x)x^{2} = y^{.}x^{2}  2x^{.}x^{2} = x^{2}y  2x^{3} 
(1  y)(1 + y + y^{2}) 
= 
(1  y)(1) + (1  y)(y) + (1  y)(y^{2}) 
 = 
1  y + y  y^{2} + y^{2}  y^{3} 
 = 
1  y^{3} 


Here are some for you...
There is quicker way of expanding expressions such as the one immediately above, called the "FOIL" method. The FOIL method says:
FOIL
 Example: 
(x + 1)(x  2) 
F 
Take the product of the First terms 
x^{.}x = x^{2} 
O 
Take the product of the Outer terms 
x^{.}(2) = 2x 
I 
Take the product of the Inner terms 
1^{.}x = x 
L 
Take the product of the Last terms 
1^{.}(2) = 2 
 Then add them all up 
x^{2}  2x + x  2 = x^{2 } x  2 

Expand the following using FOIL.
The last two examples above, and the third, are important enough to warrant special mention.
Some Identities
  Example 
Difference of two squares  (a  b)(a + b) = a^{2}  b^{2} 
(3  2x)(3 + 2x) = 9  4x^{2} 
Square of a sum  (a + b)^{2} = a^{2} + 2ab + b^{2} 
(3 + 2r)^{2} = 9 + 12r + 4r^{2} 
Square of a difference  (a  b)^{2} = a^{2}  2ab + b^{2} 
(1  3x)^{2} = 1  6x + 9x^{2} 

Some for you to try...
Now go on to the next tutorial (Part B: Factoring Algebraic Expressions) before trying the examples and exercises in Section A.3 of the Algebra Review in Applied Calculus and Finite Mathematics and Applied Calculus
Last Updated: March, 2006
Copyright © 2001 Stefan Waner and Steven R. Costenoble