Download a(b - c) = ab

•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.
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.
Multiply 67  9
6
67
9
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
60 3
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;
Multiply 67  9
6
67
9
Or use the
Distributive
Property
60 3
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
Multiply 48  7
5
48
7
Or use the
Distributive
Property
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;
33 6
Multiply 48  7
Break apart the number 48 into (50 2) – the value of this number is still
the same.
7(50  2)  350
 336
 14
Subtract
Multiply 6  473
Or use the
Distributive
Property
1
473
6
8
41
473
6
38
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(400  70  3)  2400 420
 18
283 8
Multiply 473  6
Break apart the number 473 into (400
+ 70 + 3) – the value of this number
is still the same.
 2838
Add
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
•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.
•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.
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
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