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(x+1)(x+2)
Sometimes we want to see an expression in expanded form.
x2+3x+2
This expression is the same as (x+1)(x+2). They can be written as being equal:
(x+1)(x+2) = x2+3x+2
Although the expressions are equal, they are really the same expression in different forms.
(x+1)(x+2) is factored form
x2+3x+2 is expanded form
The law of distribution, also called the distributive law, is used to convert from factored form to expanded form.
a(b+c)
ab+ac
also,
a(b+c+d)
ab+ac+ad
and
a(b+c+d+e)
ab+ac+ad+ae
The a factor outside the parenthesis has to be multiplied by every term inside the parenthesis. This is not the same as adding a number to an expression in parenthesis.For example:
a+(b+c) = a+b+c
in the case of addition the parenthesis simply go away.
When two expressions in parenthesis are multiplied together the rules are different.
(a+b)(c+d)
The first factor (a+b) is distributed to each term in the second set of parenthesis.
(b+c)
Which results in
(a+b)c+(a+b)d
Applying the distributive law again to each term results in
ac+bc+ad+bd
The steps are as follows:
| (a+b)(c+d) | factored form | |
| (a+b)c+(a+b)d | first distribution | |
| (ac+bc)+(ad+bd) | second distribution | |
| ac+bc+ad+bd | parenthesis go away, expanded form |
This method works on any factored expression
(a+b+c)(d+e+f)
(a+b+c)d+(a+b+c)e+(a+b+c)f
ad+bd+cd+ae+be+ce+af+bf+cf
Another method of expanding polynomials is called FOIL. FOIL stands for :
| F - first | |
| O - outside | |
| I - inside | |
| L - last |
(a+b)(c+d)
First: We multiply the first terms of each factor
and put the result ac
Outside: Next, the outside terms are multiplied together
the result is added to the first result ac+ad
Inside: The inside terms are multiplied together
The result is added to the previous result ac+ad+bc
Last: The last terms of both factors are multiplied together
and the result is added to the previous result ac+ad+bc+bd
A variation of the foil method works as follows
| (a+b+c)(d+e+f) |
Multiply the first term of the first factor in sequence to each term of the second factor
|
Multiply the second term of the first factor in sequence to each term of the second factor.
|
Multiply the third and last term of the first factor in seqence to each term in the second factor.
|
Another example of using these two methods includes the following:
1. First Method
| (x+1)(x+2) | factored form |
| x(x+2)+(x+2) | first distribution |
| (x2+2x)+(x+2) | second distribution |
| x2+2x+x+2 | parenthesis disappear |
| x2+3x+2 | 2x+x = 3x, like terms combine, expanded form |
Also,
|
| x4+4x3+6x2+5x+2 | expanded form |
2. FOIL
|
|
Combine like terms
| x4+4x3+6x2+5x+2 | expanded form |
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