Multiplying

The distributive property for multiplication tells us that

the Distributive Property
a(b+c) = ab+ac and (b+c)a = ba+ca

We can use this property and rules of exponents (from the previous section) to multiply out algebraic expressions involving parentheses. This is sometimes called expanding the expression.

Examples
Reading through these, stop at each step to see what happens.
  1. 3x^2(x+(1/6)x^(-3)) = (3x^2)(x) + (3x^2)((1/6)x^(-3)) = 3x^3 + (1/2)x^(-1)
  2. ((2t)^2 - 5)sqrt(t) = (2t)^2(sqrt(t)) - 5sqrt(t) = (4t^2)(t^(1/2)) - 5 t^(1/2) = 4t^(5/2) - 5t^(1/2)
  3. 2^x(3^x + 2^(x-1)) = (2^x)(3^x) + (2^x)(2^(x-1)) = ((2)(3))^x + 2^(x+x-1) = 6^x + 2^(2x - 1)

Factors with more than one term

If there are two terms in each factor, there are four terms in the product (from "FOIL"). Some of these may cancel or combine, however, leading to some special cases which appear frequently and are worth learning.

FOIL:
(a+b)(c+d) = ac + ad + bc + bd
Special Cases:
(a+b)(a-b) = a^2 - b^2
(a+b)^2 = a^2 + 2ab + b^2
(a-b)^2 = a^2 - 2ab + b^2
Examples
Again, stop at each step to see what happens.
  1. (5x^2 + 2)(x-4) = (5x^2)(x) + (5x^2)(-4) + (2)(x) + (2)(-4) = 5x^3 - 20x^2 + 2x - 8
  2. (2sqrt(r)+2)(4sqrt(r)-3) = (2)(4)(sqrt(r))^2 + (2)(-3)(sqrt(r)) + (2)(4)(sqrt(r)) + (2)(-3) = 8r + 2sqrt(r) - 6
  3. (e^x+1)(2x+e^(-x)) = (e^x)(2x) + (e^x)(e^(-x))+(1)(2x)+(1)(e^(-x) = 2xe^x + e^(x-x) + 2x + e^(-x) = 2xe^x + 1 + 2x + e^(-x)
  4. (3 - (1/2)x)^2 = 3^2 - 2(3)((1/2)x) + ((-1/2)x)^2 = 9 - 3x + (1/4)x^2
Practice

To do these practice problems make sure that you have explained each step in the examples above. Then work the practice problems until you are sure you understand them. Then go and get more practice by taking the section test at the end of this section.

Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.

  1. Expand the following into the form
    ax^p + bx^q + cx^r + d

           
    ( a x ^b + c ) ( d x ^g + h )   =   x   +   x   +   x   +  
    (Enter answers, then click: . Answer message: )
  2. Next, expand the following into the form
    a^x + b^{rx}

      ^x   ^x
    j ^k ^x ( m ^n ^x + j ^p ^x )   =     +    
    (Enter answers, then click: . Answer message: )
  3. Expand the following into the form
    a x^r + b x^s + (c x^t + d x^u)e^x

    ( s x   -   t ) ( u x ^v   +   w e^x )   =  
             
    x     +   x     +   ( x     +   ) e^x
    (Enter answers, then click: . Answer message: )
For more practice, click: .
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section test

precal: 2.1 - multiplying
page created: Mon Apr 29 00:28:25 2024
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©2003-2007 Gavin LaRose, Pat Shure / University of Michigan Math Dept. / Regents of the University of Michigan