Interactive Algebra Review

(Based on the algebra reviews in Calculus Applied to the Real World, , and Finite Mathematics and Calculus Applied to the Real World and )

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4. The Distributive Law: Multiplying Algebraic Expressions

The Distributive Law

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

Quick Examples

    x(x + 1) = x2 + x
    (y - 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
    3x(4x-1 + x4)   = 
    (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.

The last two examples, and one other, 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 + r2
Square of a difference(a - b)2 = a2 - 2ab + b2   (1 - 3x)2 = 1 - 6x + 9x2
Some for you...
(1 + x)(1 - x) =
(x + 3)2 =
(4 - x)2 =

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

Expand the following.

(x + 1)(x2 + 3x - 4)  = 
(x2 - 1/x + 1)(2x + 5)  = 
(x - 1)^3 = 

Now go over the examples and try some of the exercises in Section 4 of the Algebra Review of Calculus Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

Alternatively, press "next" button on the sidebar to go on to the next topic.

Last Updated: December, 1998
Copyright © 1998 Stefan Waner and Steven R. Costenoble