Transcript The Distributive Property The Distributive Property • The Distributive Property allows you to multiply each number inside a set of parenthesis by.

The Distributive Property
The Distributive Property
• The Distributive Property allows you to multiply each number
inside a set of parenthesis by a factor outside the parenthesis and
find the sum or difference of the resulting products.
• To distribute means to separate or break apart and then dispense
evenly.
• Sometimes it is faster and easier to break apart a multiplication
problem and use the distributive property to solve or simplify
the problem using mental math strategies.
• The distributive property is linked to factoring. When you factor
problems, you identify what numbers or variables the problem
has in common. When you distribute, you multiply the common
numbers or variables to the numbers that have been grouped
together.
The Distributive Property
When a number or letter is
separated by parentheses
and there are no other
operation symbols – it
means to distribute by
multiplying the numbers
or variables together.
For any numbers a, b, and c,
a(b + c) = ab + ac and (b + c)a = ba + bc.
a(b - c) = ab - ac and (b - c)a = ba - bc.
Notice that it doesn’t matter which side of the expression
the letter a is written on because of the symmetric property
which states for any real numbers a and b; if a = b, then b
= a.
If a(b + c) = ab + ac, then ab + ac = a(b + c).
The Distributive Property
We can use the distributive property to multiply large
numbers:
Example: 67  9
Break the number 67 into (60 + 7) & write it as 9(60 + 7).
a(b + c) = ab + ac and (b + c)a = ba + bc.
9(60  7)  540
 603
 63
The Distributive Property
Another example:
48  7
Rename the number 48 as (40 + 8) then write the
multiplication as 7(40 + 8).
a(b + c) = ab + ac and (b + c)a = ba + bc;
7(40  8)  280
 336
 56
The Distributive Property
A A three digit number can be broken apart too:
473  6
Rename the number 473 as (400 + 70 + 3).
a(b + c + d) = ab + ac + ad
6(400 70  3)  2400
 2838

420
 18
The Distributive Property
Expressions with variables:
Simplify 5(3n + 4).
No symbol between the 5 and the parenthesis indicates a
multiplication problem.
Distribute by multiplication:
5(3n  4)  5(3n )  5(4)
 15n  20
15n and 20 are not alike and therefore cannot be combined. The answer 15n + 20 is
simplified because we do not know what the value of n is at this time and cannot
complete the multiplication part of this problem.
Terms and Like Terms
Terms are either a number (constant term), a variable
(algebraic term), or a combination of numbers or variables
that are added to form an expression.
Given the problem 2x + 5, the terms are 2x and 5.
Given the problem 2x – 5, the terms are 2x and –5.
Like terms are terms that share the same variable(s) and
are raised to the same power. Remember that n’s go with
n’s n2 will only go with n2; numbers (constant term) by
themselves go with numbers by themselves.
Given 2x + 5 + 3x + 2 + 4x2 + 5x2, it is simplified as 5x + 7 + 9x2.
Equivalent Expressions
Given 5x + 4x, the expression can be simplified to
9x.
The expressions 5x + 4x and 9x are equivalent
expressions because they name the same value.
9x is now in simplest form or the expression is
said to be simplified.
Combining Like Terms
Combining like terms is the process of adding or
subtracting like terms.
Given 2x + 5 + 3x + 2 + 4x2 + 5x2,
it is simplified as 5x + 7 + 9x2.
The 2x and 3x are combined to form 5x; the 5 and
2 can be combined to form 7, and the 4x2 and 5x2
can be combined to form 9x2.
The expression is then rewritten by placing the
term with the highest exponent first, then the next
term in decreasing order. 9x2 + 5x + 7.
Coefficient
Coefficient is a number and a letter is linked
together by multiplication; the number or
numerical factor is called the coefficient.
Given the simplified algebraic expression 9x2 + 5x + 7; the
9 is the coefficient of the term 9x2, the 5 is the coefficient
of the term 5x, and the 7 is referred as the constant term.
Note: All variables have a coefficient. Given the
variable x; the coefficient is 1 because (1)(x) = x.
The expression 2x + x + x; can be simplified as
2x + 1x + 1x = 4x.
The Distributive Property
Simplify 4(7n + 2) + 6.
No symbol between the 4 and the parenthesis
indicates a multiplication problem.
4(7n  2)  6  4(7n)

4(2)
 6
 8  6
 28n  14
 28n
The constant terms 8 and 6 can be combined to form the constant number 14. The
answer 28n + 14 is simplified because we do not know what the value of n is at this
time and cannot complete the multiplication part of this problem.
The Distributive Property
Simplify 3(n + 2) + n.
No symbol between the 3 and the parenthesis
indicates a multiplication problem.
3(n  2)  n  3(n)  3(2)
 1(n)
 6  1(n)
 4n  6
 3n
Notice that n has a coefficient of 1. 3n and 1n are like terms and can be combined to
form 4n. (distributive property). The constant term 6 cannot be combined with any other
constant terms. The answer 3n + 6 is simplified because we do not know what the value
of n is at this time and cannot complete the multiplication part of this problem.