# Generating problems about rational expressions

The other day, I was solving problems regarding rational expressions. I noticed that every single time, after simplifying the numerator, the fraction is already in lowest terms. So I wondered if this is always true.

After a bit of experimentation, I was able to construct this counterexample:

$\frac{x+4}{(x+1)(x+2)} - \frac{x}{(x+2)(x+3)}$

$= \frac{(x+4)(x+3)-x(x+1)}{(x+1)(x+2)(x+3)}$

$= \frac{x^2+7x+12-x^2-x}{(x+1)(x+2)(x+3)}$

$= \frac{6x+12}{(x+1)(x+2)(x+3)}$

$= \frac{6(x+2)}{(x+1)(x+2)(x+3)}$

$= \frac{6}{(x+1)(x+3)}$.

Can you think of a way to generate infinitely-many such problems?

Here are some assumptions:

1. All the coefficients are integers.

3. After simplifying the numerator, the numerator and the denominator stilll has a common factor.

Note by Mark Lao
4 years, 7 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

Hey Mark, I am wondering what you mean by "such problems" here. Do you mean to find pairs of fractions that when added/subtracted, result in a numerator that has a common factor with the denominator? Also, the starting fractions must be linear functions of $$x$$ in the numerator? Please write back.

Staff - 4 years, 7 months ago

Do you mean to find pairs of fractions that when added/subtracted, result in a numerator that has a common factor with the denominator? - Yes.

Also, the starting fractions must be linear functions of $$x$$ in the numerator? - Not necessarily. :-)

- 4 years, 6 months ago