Transcript The Distributive Property - pams-cole

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
Distributive
Property
For any numbers a, b, and c,
a(b + c) = ab + ac and (b + c)a = ba + bc;
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.
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)
Find the sum (add) or
difference (subtract) of the
distributed products.
The Distributive Property
Multiply 67  9
6
67
9
60 3
Or use the
Distributive
Property
For any numbers a, b, and c,
now a  9, b  60, c  7
a(b + c) = ab + ac and (b + c)a = ba + bc;

 603
9(60  7)  540
Multiply 67  9
Break apart the number 67 into (60 +
7) – the value of this number is still
the same.
63
Add
a(b - c) = ab - ac and (b - c)a = ba - bc;
The Distributive Property
Multiply 67  9
6
67
9
60 3
Or use the
Distributive
Property
Multiply 67  9
Break apart the number 67 into (60 +
7) – the value of this number is still
the same.
For any numbers a, b, and c,
now a  9, b  60, c  7
a(b + c) = ab + ac and (b + c)a = ba + bc;

 603
(60  7)9  540
Add
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)
63
The Distributive Property
Multiply 48  7
5
48
7
33 6
Or use the
Distributive
Property
Multiply 48  7
Break apart the number 48 into (50 2) – the value of this number is still
the same.
For any numbers a, b, and c,
now a  7, b  50, c  2
a(b + c) = ab + ac and (b + c)a = ba + bc;
a(b - c) = ab - ac and (b - c)a = ba - bc;
7(50  2)  350
 336
 14
Subtract
The Distributive Property
Multiply 6  473
Or use the
Distributive
Property
1
473
6
8
41
473
For any numbers a, b, c, and d
now a  6, b  400, c  70, d  3
a(b + c + d) = ab + ac + ad
41
473
6
6
38
283 8
Multiply 473  6
Break apart the number 473 into (400
+ 70 + 3) – the value of this number
is still the same.
6(400  70  3)  2400 420
 2838
 18
Add
The Distributive Property
Simplify 5(3n + 4)
Notice the pattern:
No symbol between the 5 and the parenthesis indicates a multiplication problem.
Distribute by multiplication then perform the indicated operation inside the parenthesis.
Use the
Distributive
Property
Notice that 15n means (15)(n) and is linked
by multiplication and that the number 20 is
by itself. These two terms 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.
For any numbers a, b, and c
now a  5, b  3n, c  4
a(b + c ) = ab + ac
5(3n  4)  5(3n )
 15n
Simplified


5( 4)
20
The Distributive Property
• Term – 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 ; x’s
go with x’s; n2 will only go with n2; numbers (constant term) by
themselves go with numbers by themselves. Given the problem
2x + 5 + 3x + 2 + 4x2 + 5x2 can be simplified as 5x + 7 + 9x2.
• Equivalent expression – Given the problem 5x + 4x; 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.
The Distributive Property
• Combining like terms – the process of adding or subtracting like terms.
Given the problem 2x + 5 + 3x + 2 + 4x2 + 5x2 can be simplified as 5x
+ 7 + 9x2. The 2x and 3x can be 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 simplified problem is then rewritten by placing the term with
the highest exponent first, then the next term in decreasing order. 9x2 +
5x + 7
• Coefficient –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. Given the problem 2x + x + x; can
be simplified as 2x + 1x + 1x = 4x.
The Distributive Property
Simplify 4(7n + 2) +6
Notice the pattern:
No symbol between the 4 and the parenthesis indicates a multiplication problem.
Distribute by multiplication then perform the indicated operation inside the parenthesis.
Use the
Distributive
Property
Notice that 28n cannot be combined with
any other n terms. The constant terms 8 and
6 are linked with addition and 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.
For any numbers a, b, and c
a(b + c ) = ab + ac
4(7n  2)  6  4(7 n)
 28n
Simplified
 28n



4( 2)
8
14


6
6
The Distributive Property
Simplify 3(n + 2) + n
Notice the pattern:
No symbol between the 3 and the parenthesis indicates a multiplication problem.
Distribute by multiplication then perform the indicated operation inside the parenthesis.
Use the
Distributive
Property
Notice that n has a coefficient of 1. After
applying the distributive property – you can
combine like terms. 3n and 1n can be
combined to form 4n. 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.
For any numbers a, b, and c
a(b + c ) = ab + ac
3(n  2)  n  3( n )
 3n
Simplified

4n



1(n)
 1(n)
3( 2)
6
6