Transcript distributive property

The
Distributive
Property
To use the distributive property to simplify
expressions.
Look at this problem:
2(4 + 3)
Through your knowledge of order of operations, you know what to do
first to evaluate this expression.
2(7)
14
Now, look what happens when I do something different with the
problem.
*
*
2(4 + 3) = 8 + 6 = 14
No difference.
This is an example of the
distributive property.
© William James Calhoun
Now why would one ever use the distributive property to solve 2(4 + 3)?
The answer is generally,
“Never! Just use the order of operations.”
Where this is going to become very important is when we have an
expression in the parenthesis which can not be simplified, like:
2(4 + x)
You will use the distributive property throughout all of Algebra.
© William James Calhoun
DISTRIBUTIVE PROPERTY
For any numbers a, b, and c,
a(b + c) = ab + ac and (b + c)a = ba + ca;
a(b - c) = ab - ac and (b - c)a = ba - ca.
Another way to think of it is, “When multiplying into parenthesis,
everything on the inside gets a piece of what is on the outside.”
Notice the number to be distributed can either be at the front of the
parenthesis or at the back.
If there is no number visible in front or in back of the parenthesis, the
number to be distributed is 1.
© William James Calhoun
Here are a couple of definitions that will be used a great deal.
• TERM - number, variable, or product or quotient of numbers
and variables
Examples of terms:
x3, 1/4a, and 4y.
The expression 9y2 + 13y + 3 has three terms.
TERMS
• LIKE TERMS - terms that contain the same variables, with
corresponding variables having the same power
© William James Calhoun
Terms must have the EXACT same letters to the EXACT same powers
in order to be LIKE terms!
8x2 + 2x2 + 5a + a
8x2 and 2x2 are like terms.
5a and a are also like terms.
Another way to think of it is this:
Like terms are alike in that they have the exact same letter configuration.
8x and 4x2 are not like terms because they do not have the same power.
© William James Calhoun
ONLY LIKE TERMS CAN BE COMBINED THROUGH
ADDITION AND SUBTRACTION.
Since 3x and 8x are like terms, they can combine - both have the same
letter configuration - an “x” to the 1st power.
We can use the distributive property to undistribute the x and combine
the numbers:
3x + 8x
= (3 + 8)x
= 11x
Another way to look at the problem is, “You have three x’s plus eight x’s.
All told, how many x’s do you have?”
The answer is, “You have eleven x’s,” or just:
11x.
© William James Calhoun
EX2β
To simplify an expression in math, you must:
1) Have all like terms combined; and
2) Have NO parenthesis are present.
1 2
11 2
2
EXAMPLE 2α: Simplify x  2x  x .
4
4
In this expression,
1 2
11
x , 2x 2 , and x 2 are all like terms.
4
4
1 2
11
x  2x 2  x 2
4
4
Undistribute the x2.
11 
1
   2   x2
4
4
Add the numbers up.
= (5)x2
Final answer.
= 5x2
© William James Calhoun
• coefficient - the number in front of the letters in a term
In the term 23ab, 23 is the coefficient.
In xy, the coefficient is 1.
NEVER FORGET THE “INVISIBLE” ONE!
EXAMPLE: Name the coefficient in each term.
a. 145x2y
b. ab2
145
1
EXAMPLE: Name the coefficient in each term.
a. y2
b.
r
4
4a 2
c.
5
4/
5
c.
5x 2
7
© William James Calhoun
EXAMPLE 4α: Simplify each expression.
3
a
4
4
2
2
a. 4w + w + 3w - 2w
b.
 2a 3
4
= (4 + 1)w4 + (3 - 2)w2
= (1/4 + 2)a3
= 5w4 + 1w2
= 21/4a3
= 5w4 + w2
EXAMPLE: Simplify each expression.
a. 13a2 + 8a2 + 6b
b. 5m  m  4m 2
6
6
© William James Calhoun