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:

All the coefficients are integers.
The addends are already in lowest terms.
After simplifying the numerator, the numerator and the denominator stilll has a common factor.

#Algebra #ProblemWriting #RationalExpression

Easy Math Editor

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`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`	$\sum_{i=1}^3$
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Comments

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.

Josh Silverman Staff - 6 years, 10 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. :-)

Mark Lao - 6 years, 9 months ago

Math	Appears 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}`	$\boxed{123}$