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Example:Simplify Polynomial Expression

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Distributive property

In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra. For example:

2 × (1 + 3) = (2 × 1) + (2 × 3).

4(8+2) = (4×8) + (4×2) because

4(8 + 2) = 4(10) = 40

(4×8) + (4×2) = 32 + 8 = 40

In the left-hand side of the above equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the results added afterwards. Because these give the same final answer (8), we say that multiplication by 2 distributes over addition of 1 and 3. Since we could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers.

Definition

Given a set S and two binary operations · and + on S, we say that the operation ·

1.       is left-distributive over + if, given any elements x, y, and z of S,

x · (y + z) = (x · y) + (x · z); (Our solved example in mathguru.com uses this concept)

2.       is right-distributive over + if, given any elements x, y, and z of S:

(y + z) · x = (y · x) + (z · x);

3.       is distributive over + if it is both left- and right-distributive.

http://en.wikipedia.org/wiki/Distributive_property

Commutative property

In mathematics an operation is commutative if changing the order of the operands does not change the end result. (Our solved example in mathguru.com uses this concept)

It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. By contrast, division and subtraction are not commutative.

Mathematical definition

A binary operation  on a set S is said to be commutative if: (Our solved example in mathguru.com uses this concept)

An operation that does not satisfy the above property is called non commutative.

http://en.wikipedia.org/wiki/Commutative_property

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.