distribute to expand, simplify, or factor
Intro
It is not necessary to
distribute
until
variables
are used in algebra.
On the left, arithmetic syntax of
order of operations
is used.
On the right, the algebraic techniques of the distributive property is used.
arithmetic algebra 2(4 + 5) = 2(4 + 5) 2(9) = 2(4)+2(5) 18 = 8 + 10 18 = 18
The arithmetic use of
inclusion marks
or
parentheses
is easier.
You try it.
If you'd like,
1st: Mentally compute the value using order of operations.
2nd: Use the distributive property to simplify 3( 4 + 5 - 6 - 3)
Answer 3( 4 + 5 - 6 - 3) 3(4) + 3(5) - 3(6) - 3(3) 12 + 15 - 18 - 9 0
So, Why Is The Distributive Property Needed?
The essay
Most Operations Are Not Distributive
gives examples
to answer that question.
In short, you need it to expand and factor expressions which are
the sum or difference of variable and constant terms.
One use undoes the other.
Distribute to expand:
3(x + 4)
Answer 3(x + 4) 3(x) + 3(4) 3x + 12
Distribute to factor:
3x + 12
Answer 3x + 12 3(x) + 3(4) 3(x + 4)
The Distributive Property in Use
I. Simplify expressions 1 and 2 then check each answer.
1.] 3(x - 5) - 2(x + 3 - 2y)
Answer x +4y - 21
2.] -5
^{2}
- 4(x - y) + 4
^{0}
- 0
^{4}
Answer -4x + 4y -24
II. Solve equations 3 which uses the expressions from problems 1 and 2.
3.] 3(x - 5) - 2(x + 3 - 2y) = -5
^{2}
- 4(x - y) + 4
^{0}
- 0
^{4}
Answer x + 4y - 21 = - 4x + 4y - 24 - 4y = - 4y ---------------------------- x - 21 = - 4x - 24 +4x = + 4x ------------------------------ 5x - 21 = -24 5x - 21 = - 24 + 21 + 21 --------------- 5x = -3 5x/5 = -3/5 x = -3/5
III. Simplify expressions 4 and 5 then check each answer.
The
caret
, ^ , is used to indicate exponentiation.
4.] 5x(x
^{2}
+ 4x - 3) - (x
^{3}
- 4x
^{2}
- 4)
Answer 5x(x^2 + 4x - 3) - (x^3 - 4x^2 - 4) 5x^3 + 20x^2 - 15x - (x^3 - 4x^2 - 4) 5x^3 + 20x^2 - 15x - x^3 + 4x^2 + 4 5x^3 - x^3 + 20x^2 + 4x^2 - 15x + 4 4x^3 + 24x^2 - 15x + 4
5.] 3(6 - 7)
^{3}
- 2
^{2}
(3 - ( 5 - 1))
Answer 3(6 - 7)^3 - 2^2(3 - ( 5 - 1)) 3(-1)^3 - 4(3 - ( 4)) 3(-1) - 4(3 - 4) 3(-1) - 4(-1) -3 + 4 1
IV. Solve equation 6 which uses the expressions from problems 4 and 5 and other terms.
6.] 5x(x
^{2}
+ 4x - 3) - (x
^{3}
- 4x
^{2}
- 4) = {3(6 - 7)
^{3}
- 2
^{2}
(3 - ( 5 - 1)) } + 4x
^{3}
+ 24x
^{2}
Answer (See work above.) 5x(x^2+4x-3)-(x^3-4x^2-4)={3(6-7)^3-2^2(3-(5-1))}+4x^3+24x^2 4x^3 + 24x^2 - 15x + 4 = 1 + 4x^3 + 24x^2 4x^3 + 24x^2 - 15x + 4 = 1 + 4x^3 + 24x^2 -4x^3 -4x^3 -------------------------------------------- + 24x^2 - 15x + 4 = 1 + 24x^2 - 24x^2 - 24x^2 -------------------------------------------- - 15x + 4 = 1 - 15x + 4 = 1 - 4 -4 -------------- - 15x = -3 - 15x/(-15)= -3/(-15) x = 1/5
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