Question about order of operations when simplifying fractions

[Derpman]

x(x + x + m + b)
----------------
x

x(2x + m + b)
---------------
x

Ok, so at this point why wouldn't one distribute the X before dividing by X? I know if one does that, then they will get the incorrect answer. The correct way to do this problem is to divide the X (cancel them), leaving 2x + m + b. But doesn't the order of operations call for one to get rid of the parenthesis first by distributing the X?

Thank you.

 

[tkhunny]

Well, the order of operations demands only that you settle arguments. It does not mandate the order in which you must proceed.

The joy of mathematics, generally, I that unique results do NOT care how you find tem. If you violate no principle along the way, the result should be the same.

The Division First, as you suggested.
[x * (2x + m + b)] / x = (x/x) * (2x + m + b) = 1 * (2x + m + b) = 2x + m + b

The Distributive Property of Multiplication Over Addition First.
[x * (2x + m + b)] / x = (2x^2 + mx + bx)/x = (2x^2 / x) + (mx / x) + (bx / x) = 2x(x/x) + m(x/x) + b(x/x)= 2x * 1 + m * 1 + b * 1 = 2x + m + b

Same results.

 

[mmm4444bot]

The Order of Operations mostly pertains to evaluating expressions. When we simplify expressions or solve equations, the Order of Operations does not always apply.

Also, the grouping-symbols step in the Order of Operations (i.e., parentheses, and other types of grouping symbols) doesn't say that you need to use distribution; it says that you need to work inside the grouping symbols first. You did that.

x(x+x+m+b)/x

x(2x + m + b)/x

Now, since there are no exponentiations, the next step in the Order of Operations says to do multiplications and divisions before any additions or subtractions.

You can divide x by x first. You're not required to do the multiplication first.

2x + m + b :cool:

 

[JeffM]

x(x + x + m + b)
----------------
x

x(2x + m + b)
---------------
x

Ok, so at this point why wouldn't one distribute the X before dividing by X? I know if one does that, then they will get the incorrect answer.

No. You will get the correct answer.

\(\displaystyle \dfrac{x(2x + m + b)}{x} = \dfrac{2x^2}{x} + \dfrac{mx}{x} + \dfrac{bx}{x} = 2x + m + b.\)

 

[Derpman]

Oh. Well that was embarrassing.

Thanks for the responses!

 

[mmm4444bot]

Well that was embarrassing.

We've all been there! :cool:

A little embarrassment is a good thing; when the brain detects a self-contradiction, it rewires itself to avoid such contradiction in the future. When there is a strong emotional response associated with the situation, the brain grows even more so.