1. Divide polynomials

So, I have examples in my math book, and it says study the examples until you understand how the answer was obtained. This one confuses me:

$\displaystyle \frac{-5a^2bc}{25ab^2c^2}=-\frac{a}{5bc}$

Looking at the answer, I have these questions:

1. Why does 5 end on the bottom of the fraction bar? Why does 5 exist anyway? The answer to -5/25 is -0.2, so why not use that number instead of 5?

2. Why is the negative sign on the outside of the fraction?

3. Why does $\displaystyle a$ end on top and $\displaystyle bc$ on bottom?

I'm thinking the answer should be: $\displaystyle \frac{-0.2a}{bc}$ but that is not the answer. What am I doing wrong? What is the step by step process for dividing polynomials?

2. Originally Posted by bryang
So, I have examples in my math book, and it says study the examples until you understand how the answer was obtained. This one confuses me:

$\displaystyle \frac{-5a^2bc}{25ab^2c^2}=-\frac{a}{5bc}$

Looking at the answer, I have these questions:

1. Why does 5 end on the bottom of the fraction bar? Why does 5 exist anyway? The answer to -5/25 is -0.2, so why not use that number instead of 5?

The 5 ends up on the bottom because your original expression has a 25 on the bottom and a 5 on the top. $\displaystyle \frac{5}{25}$. If you take out a factor of 5 on both sides of the fraction you get $\displaystyle \frac{5(1)}{5(5)}$. The five on the top will cancel with the five on the bottom, leaving just $\displaystyle \frac{1}{5}$. The reason we don't write it in decimal form is that fractions are simpler to manipulate than decimals. For example, if I asked you to multiply your equation by a third, if you did it with decimals, you'd end up with an infinite decimal (since a third i 0.333333333333333333333333333.... recurring!). But if you multiplied it by the fraction $\displaystyle \frac{1}{3}$, there would be no infinite fractions . In general, use fractions instead of demicals, they look nicer!

2. Why is the negative sign on the outside of the fraction?

It doesn't really matter if you put it inside or outside. It means the same thing.

3. Why does $\displaystyle a$ end on top and $\displaystyle bc$ on bottom?

$\displaystyle a^2$ on top, and $\displaystyle a$ on the bottom! And the same kind of thing.

I'm thinking the answer should be: $\displaystyle \frac{-0.2a}{bc}$ but that is not the answer. What am I doing wrong? What is the step by step process for dividing polynomials?

You aren't doing anything wrong. Your answer is correct. The only difference is that they have written 0.2 as a fraction, and you have written it as a decimal.
Mush

3. Originally Posted by bryang
So, I have examples in my math book, and it says study the examples until you understand how the answer was obtained. This one confuses me:

$\displaystyle \frac{-5a^2bc}{25ab^2c^2}=-\frac{a}{5bc}$

Looking at the answer, I have these questions:

I'm thinking the answer should be: $\displaystyle \frac{-0.2a}{bc}$ but that is not the answer. What am I doing wrong? What is the step by step process for dividing polynomials?
Hello bryang,

1. Why does 5 end on the bottom of the fraction bar? Why does 5 exist anyway? The answer to -5/25 is -0.2, so why not use that number instead of 5?
$\displaystyle \frac{-5a^2bc}{25ab^2c^2}=-\frac{a}{5bc}$

The 5 and 25 have a common factor of 5, so the 5 in the numerator divided into the denominator 25 five times. Your answer of -.2 is correct because the fraction became $\displaystyle \frac{-1a^2bc}{5ab^2c^2}$, and $\displaystyle -\frac{1}{5}$ is the same as -.2; but we don't want to complicate things by mixing fractions and decimals.

2. Why is the negative sign on the outside of the fraction?
Where would you like it to be? If a fraction has a negative value, you may assign the negative sign to either the numerator or denominator or to the fraction itself.

$\displaystyle -\frac{a}{5bc}=\frac{-a}{5bc}=\frac{a}{-5bc}$

3. Why does $\displaystyle a$ end on top and $\displaystyle bc$ on bottom?
Here, again, it has to do with common factors in the numerator and denominator. $\displaystyle a$ in the denominator factors $\displaystyle a^2$ in the numerator $\displaystyle a$ times. Likewise, $\displaystyle bc$ in the numerator factors $\displaystyle b^2c^2$ in the denominator $\displaystyle bc$ times.

There's a longer explanation regarding laws of exponents if you're ready for that.

I see that Mush beat me to the answer, but I'm sending it anyway.