Problems changing my whole outlook on math

[A.J.710]

I have been doing some problems today on my online math course and the preferred way of doing it is changing the whole concept of fractions I've learned my entire life.

Normally I would go about doing numbers individually in a polynomial for example 25a^2/25a would be just a then move onto the next set of numbers.

Now the course wants me to factor the entire numerator and factor the entire denominator and then simplify.

I'd just like some explanation behind this because I was never taught to do this before and it just has me a little confused.

 

[Ray Vickson]

I have been doing some problems today on my online math course and the preferred way of doing it is changing the whole concept of fractions I've learned my entire life.

Normally I would go about doing numbers individually in a polynomial for example 25a^2/25a would be just a then move onto the next set of numbers.

Now the course wants me to factor the entire numerator and factor the entire denominator and then simplify.

I'd just like some explanation behind this because I was never taught to do this before and it just has me a little confused.

If you are thinking of an example like
\frac{x^2 - 1}{x-1} = x+1
then you can see that you just cannot look at the numerator $x^2 - 1$ term-by-term; to get anywhere, you need to look at the thing in its entirety. What is happening is that we can write $x^2 - 1 = (x-1)(x+1)$, and then we can cancel out the $x-1$ factors from both the numerator and denominator.

There are several aspects involved in learning such material. (1) You need to practice---do lots of examples. (2) Understanding comes through doing, so even if you are unsure initially, you can just go ahead and apply the steps without, maybe, understanding WHY they work. After you have seen several instances you should be in a better position to appreciate the procedure and what it can do for you.

Here is another example for you to try (with broad hints provided):
\text{simplify the fraction } \frac{2x^2 + 6x -20}{x^2 +4x -12}.
Note that the numerator can be written as $2x^2 + 6x - 20 = 2(x-2)(x+5)$ and the denominator can be written as $x^2 + 4x - 12 = (x-2)(x+6)$.

 

[A.J.710]

If you are thinking of an example like
\frac{x^2 - 1}{x-1} = x+1
then you can see that you just cannot look at the numerator $x^2 - 1$ term-by-term; to get anywhere, you need to look at the thing in its entirety. What is happening is that we can write $x^2 - 1 = (x-1)(x+1)$, and then we can cancel out the $x-1$ factors from both the numerator and denominator.

There are several aspects involved in learning such material. (1) You need to practice---do lots of examples. (2) Understanding comes through doing, so even if you are unsure initially, you can just go ahead and apply the steps without, maybe, understanding WHY they work. After you have seen several instances you should be in a better position to appreciate the procedure and what it can do for you.

Here is another example for you to try (with broad hints provided):
\text{simplify the fraction } \frac{2x^2 + 6x -20}{x^2 +4x -12}.
Note that the numerator can be written as $2x^2 + 6x - 20 = 2(x-2)(x+5)$ and the denominator can be written as $x^2 + 4x - 12 = (x-2)(x+6)$.

Ok I'll keep practicing, but as for the problem you just gave me. would that simplify down to

2x + 10/x + 6

 

[Mark44]

Ok I'll keep practicing, but as for the problem you just gave me. would that simplify down to

2x + 10/x + 6

What you wrote is different from what you meant.

This is what you wrote:
$$2x + \frac {10} x + 6$$

To indicate that 2x + 10 is the numerator and that x + 6 is the denominator, you have to use parentheses, like this: (2x + 10)/(x + 6).

 

[A.J.710]

What you wrote is different from what you meant.

This is what you wrote:
$$2x + \frac {10} x + 6$$

To indicate that 2x + 10 is the numerator and that x + 6 is the denominator, you have to use parentheses, like this: (2x + 10)/(x + 6).

Sorry about that. I'm not used to writing math on a computer like this.

 

[SteamKing]

It helps tremendously if you use parentheses to clarify expressions:

(2x+10)/(x+6) rather than 2x + 10/x + 6, which might be read as "2 times x plus 10 over x plus 6"

 

[Mark44]

In addition, what you wrote as 25a²/25a also needs parentheses. Without them, it would be interpreted as (25a²/25) * a.

The correct and unambiguous way to write this expression is 25a²/(25a).

 

[Ray Vickson]

Ok I'll keep practicing, but as for the problem you just gave me. would that simplify down to

2x + 10/x + 6

You have the right idea, but your answer is badly wriiten. What you have written means 2x + \frac{10}{x} + 6 (when read and parsed using standard rules). If you mean to write
\frac{2x + 10}{x+6} then you need to use parentheses, like this: (2x+10)/(x+6). When written like that it would be 100% correct.

 

[A.J.710]

How do you guys do the fraction thing? So it looks like it's on paper?

 

[micromass]

How do you guys do the fraction thing? So it looks like it's on paper?

See https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

In particular:

##\frac{2x^2 + 1}{x+1}##

Gives: $\frac{2x^2 + 1}{x+1}$.

$$\frac{2x^2 + 1}{x+1}$$

Gives: $$\frac{2x^2 + 1}{x+1}$$ and

 

[Mark44]

How do you guys do the fraction thing? So it looks like it's on paper?

We're using LaTeX code.

You can see what we did by right-clicking the stuff in LaTeX. Here's a link to a PF topic on how to write using LaTeX - https://www.physicsforums.com/showpost.php?p=3977517&postcount=3.

 

[symbolipoint]

A.J.710 wrote this:

I'd just like some explanation behind this because I was never taught to do this before and it just has me a little confused.

You are now supposed to learn MORE than what you were taught before. The numbers and expressions that you will be given to solve in your instruction and also in real life can have repeated factors in the number or numbers which you will work with and simplify.