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.

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.

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.

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.

The Distributive Property

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).

a(b + c) = ab + ac and (b + c)a = ba + bc.a(b + c) = ab + ac and (b + c)a = ba + bc.

=+ )760(9 540 + 63

603=

The Distributive PropertyWe 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;a(b + c) = ab + ac and (b + c)a = ba + bc;

7(40+8)= 280+ 56

=336

The Distributive PropertyAnother example:

48 7

Rename the number 48 as (40 + 8) then write the multiplication as 7(40 + 8).

a(b + c + d) = ab + ac + ada(b + c + d) = ab + ac + ad

=++ )370400(6 2400 + 420

2838=

+ 18

The Distributive PropertyA A three digit number can be broken apart too:

473 6Rename the number 473 as (400 + 70 + 3).

Distribute by multiplication:Distribute by multiplication:

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.

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.

=+ )43(5 n )3(5 n

+

)4(5+

= n15 20

The Distributive PropertyExpressions with variables:

Simplify 5(3n + 4).No symbol between the 5 and the parenthesis indicates a multiplication problem.

Terms and Like TermsTerms 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 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 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.

=++ 6)27(4 n )7(4 n

+

)2(4+

= n28 8

+ 6

+ 6

= n28 + 14

The Distributive PropertySimplify 4(7n + 2) + 6.

No symbol between the 4 and the parenthesis indicates a multiplication problem.

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.

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.

=++ nn )2(3 )(3 n

+)2(3+

= n3 6+ )(1 n

+ )(1 n

= n4 + 6

The Distributive PropertySimplify 3(n + 2) + n.

No symbol between the 3 and the parenthesis indicates a multiplication problem.