Polynomial fractions simplest form?

[bonodut]

I was taught that when you have a polynomial fraction where the denominator is of a higher degree than the numerator, it can't be reduced any further. This seems wrong to me for a couple of reasons.

1. If the denominator can be factored some of the terms may cancel out

2. Say you have the expressions

a)

b)

B divides out to

Couldn't A be expressed as the reciprocal of this? If so, which is the simplest form? A or this reciprocal? Why?

I don't think I understand what "simplest form" means in terms of polynomial fractions. Hopefully this question makes sense.

 

[PeroK]

If the denominator is a higher power than the numerator, you can't do polynomial long division. But, as you have shown, you may be able to simplify by cancelling common factors.

I wouldn't worry unduly about "simplest form". If you get a more complicated expression, there's no clear definition of what counts as "simplest".

You might prefer $1 - \frac{1}{1+x}$ and I might prefer $\frac{x}{1+x}$, for example.

 

[Mentallic]

B divides out to View attachment 79327

Couldn't A be expressed as the reciprocal of this? If so, which is the simplest form? A or this reciprocal? Why?

It can be the reciprocal of that, which would be

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

but even though "simplest form" is a subjective term at times, fractions within fractions are hardly ever considered most simple. Only in rare circumstances that don't apply here.

 

[symbolipoint]

Your listed items labeled a and b are not polynomials.

 

[bonodut]

Your listed items labeled a and b are not polynomials.

I never said they were. I think it was pretty clear I was asking about ratios of polynomials.

 

[Mentallic]

I never said they were. I think it was pretty clear I was asking about ratios of polynomials.

A polynomial divided by another polynomial is quite common so they've been given the name "rational functions".

 

[symbolipoint]

I never said they were. I think it was pretty clear I was asking about ratios of polynomials.

Those expressions are not ratios of polynomials either. Each of the two expressions has ONE rational expression each combined with the other terms by addition.
Item a contains 1/x^2, and item b contains 3/x.

 

[symbolipoint]

Those expressions are not ratios of polynomials either. Each of the two expressions has ONE rational expression each combined with the other terms by addition.
Item a contains 1/x^2, and item b contains 3/x.

bonodut,
I now see part of what you meant. "polynomial ratios", should be called "rational expressions". Did you maybe not supply all the needed grouping symbols for unambiguous reading, or to show exactly what your items a and b are?

 

[symbolipoint]

I was taught that when you have a polynomial fraction where the denominator is of a higher degree than the numerator, it can't be reduced any further. This seems wrong to me for a couple of reasons.

1. If the denominator can be factored some of the terms may cancel out

2. Say you have the expressions

a) View attachment 79325

b) View attachment 79324

B divides out to View attachment 79327

Couldn't A be expressed as the reciprocal of this? If so, which is the simplest form? A or this reciprocal? Why?

I don't think I understand what "simplest form" means in terms of polynomial fractions. Hopefully this question makes sense.

Do you mean for item a,
(x+1)/(x^2+2x+3)

and do you want to simplify that?
The degree of denominator is higher than degree of numerator. Performing Division is not too useful. Can you factorize the denominator? No. Even if the denominator here were factorable, if you could perform a cancelation with a possible x+1 binomial factor, this would change the rational expression's meaning. This rational expression, not simplifiable.

 

[bonodut]

bonodut,
I now see part of what you meant. "polynomial ratios", should be called "rational expressions". Did you maybe not supply all the needed grouping symbols for unambiguous reading, or to show exactly what your items a and b are?

Yes, I see now how it

Do you mean for item a,
(x+1)/(x^2+2x+3)

and do you want to simplify that?
The degree of denominator is higher than degree of numerator. Performing Division is not too useful. Can you factorize the denominator? No. Even if the denominator here were factorable, if you could perform a cancelation with a possible x+1 binomial factor, this would change the rational expression's meaning. This rational expression, not simplifiable.

Yes, I see how it was confusing the way I wrote it (used a "latex" generator and couldn't figure out how to do a horizontal line for the ratio).

Thanks for your answer. It's so obvious now. It's analogous to why you could "simplify" 11/5 by turning it into a mixed number but can't do anything to 5/11

 

[symbolipoint]

$$(x+1)/(x^2+2x+3)$$
$$\frac{x+1}{x^2+2x+3}$$
just testing LaTeX or TeXRight-click the expression and choose... TeX commands.
A small window will open, showing the code which gives the expression.
Not shown in that code is the left and right tags used. They are on the left: Left square bracket, t, e, x, right square bracket. On the right: Left square bracket, backslash, t, e, x, right square bracket.