Interactive Algebra Review

Based on the algebra reviews in Applied Calculus and Finite Mathematics and Applied Calculus

For best viewing, adjust the window width to at least the length of the line below.

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 = x2 + x
    (y - 2x)x2 = y.x2 - 2x.x2 = x2y - 2x3
    (1 - y)(1 + y + y2) = (1 - y)(1) + (1 - y)(y) + (1 - y)(y2)
    = 1 - y + y - y2 + y2 - y3
    = 1 - y3

Here are some for you...

Q1 3x(4 - x4)   = 
Q2 3x(4x-1 + xy)   = 
Q3 -3x(4x2 - 5x + 2)  = 
Q4 (x + xy -y)xy  = 
Q5 -x(x2 + 2xy - y2)y  = 
Q6 (2x + 3)(4x - 5)  = 

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 = x2
    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 x2 - 2x + x - 2  =  x2 - x - 2

 

Expand the following using FOIL.

Q1 (4x - 3)(2x + 5)   = 
Q2 (-x + 2)(2x - 1)   = 
Q3 (x + 3)2   = 
Q4 (2x2 - x)(2x + 4)   = 
Q5 (a - b)(a + b)   = 
Q6 (x - y)2   = 

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) = a2 - b2       (3 - 2x)(3 + 2x) = 9 - 4x2
Square of a sum(a + b)2 = a2 + 2ab + b2       (3 + 2r)2 = 9 + 12r + 4r2
Square of a difference(a - b)2 = a2 - 2ab + b2       (1 - 3x)2 = 1 - 6x + 9x2

Some for you to try...

Q1 (1 + x)(1 - x) =
Q2 (x - 3)(x + 3) =
Q3 (3x2 - y)( 3x2 + y) =
Q4 (x + 3)2 =
Q5 (4 - x)2 =
Q6 (-x + 3y)2 =

 
To wind up this unit, here are some miscellaneaous exercises for you to try.

Expand the following.

Q7 (x + 1)(x2 + 3x - 4)  = 
Q8 (x2 - 1/x + 1)(2x + 5)  = 
Q9 (x - 1)3  =  
Q10 (x - y)(x2 + xy + y2)  =  

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