# Does This Equation Hold in All Commutative Rings?

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• donglepuss
In summary, the equation ##(a^3+x)(b^2-y)=a^3b^2-a^3y+xb^2-xy## is a special kind of identity that is true for all values of the variables, including whole numbers, integers, rationals, reals, and complex numbers. It also holds for square matrices and other mathematical structures that support multiplication distributing over addition, assuming commutativity.
donglepuss
TL;DR Summary
##(a^3+x)(b^2-y)=a^3(b^2)-ya^3+xb^2-xy##
##(a^3+x)(b^2-y)=a^3(b^2)-ya^3+xb^2-xy##

is this correct for all whole numbers x,y,a,b?

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Yes. Even for rational or real numbers.

donglepuss said:
is this correct for all whole numbers x,y,a,b?
##(a^3+x)(b^2-y)=a^3(b^2)-ya^3+xb^2-xy##
This equation is a special kind of equation: an identity, one that is true for all values of the variables, whether whole numbers, integers, rationals, reals, or complex numbers. It's even true for square matrices, as long as they are all the same size.

A simpler example of an identity is this: ##a(b + c) = ab + ac##, which is true for any mathematical structures that support multiplication distributing over addition.

jedishrfu
Your equation isn't written in its best form. If you compare @Mark44's formula with what you have written, then
donglepuss said:
##(a^3+x)(b^2-y)=a^3(b^2)-ya^3+xb^2-xy##
is a bit inaccurate. There are structures in mathematics which are in general not commutative, rings and algebras. So it is a good habit to learn it by respecting left and right, so it's better to write
##(a^3+x)(b^2-y)=a^3b^2-a^3y+xb^2-xy##

jedishrfu
fresh_42 said:
is a bit inaccurate.
I didn't notice that ##a^3## times ##-y## was written as ##-ya^3##, when I mentioned matrix multiplication, which isn't generally commutative.

jedishrfu
Notice how the general binomial ##(a+b)^n = \Sigma _{i=0}^n (nCi )x^n y^i ## ; ##nCi## := " n choose i" also assumes commutativity. I think the OP and this identity hold for all commutative rings.

## 1. What are the steps to check if an equation is correct?

The steps to check if an equation is correct are:1. Rewrite the equation in standard form.2. Simplify each side of the equation by combining like terms.3. Check if the equation is balanced by ensuring that both sides have the same value.4. Substitute values for the variables and solve the equation.5. Check if the solution satisfies the original equation.

## 2. How do I know if an equation is balanced?

An equation is balanced when both sides have the same value. This means that the number of atoms or molecules on each side of the equation is equal.

## 3. What are the common mistakes to look out for when checking an equation?

Some common mistakes to look out for when checking an equation are:1. Incorrectly applying the order of operations.2. Forgetting to distribute or combine like terms.3. Making a sign error, such as forgetting to change the sign when multiplying or dividing by a negative number.4. Incorrectly solving for a variable.5. Forgetting to check the solution in the original equation.

## 4. Can I use a calculator to check if an equation is correct?

Yes, you can use a calculator to check if an equation is correct. However, it is important to be familiar with the steps to check an equation manually in case the calculator gives an incorrect answer.

## 5. What should I do if the equation is not correct?

If the equation is not correct, you should carefully review your steps and check for any mistakes. If you are unable to find the error, it is best to seek help from a teacher or tutor. It is also important to practice solving equations regularly to improve your skills and avoid making mistakes in the future.

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