Question about order of operations when simplifying fractions

Derpman

New member
Joined
Apr 28, 2017
Messages
2
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.
 
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.
 
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:
 
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.

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