Simplify (Adding and Subtracting Rational Functions)

In summary, to simplify the given expression, we need to find the lowest common denominator of 60y^4 and then express each term with that denominator. After simplifying, the resulting expression is $\frac{20xy^3-5x^2y+9}{60y^4}$.
  • #1
eleventhxhour
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5c) Simplify.

\(\displaystyle \frac{2x}{3y} - \frac{x^2}{4y^3} + \frac{3}{5y^4} \)This is what I did, which is wrong according to the textbook. Could someone point out what I did wrong and how to correct it? Thanks.

\(\displaystyle \frac{(2x)(4y^3)-(x^2)(3y)}{(3y)(4y^3)} + \frac{3}{5y^4}\)

\(\displaystyle \frac{8xy^3-3x^2y}{(12y^3)} + \frac{3}{5y^4}\)

\(\displaystyle \frac{(8xy^3-3x^2y)(5y^4)+(3)(12y^4)}{(12y^3)(5y^4)}\)

\(\displaystyle \frac{(40xy^7-15x^2y^5)+36y^4}{60y^8}\)
 
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  • #2
eleventhxhour said:
5c) Simplify.

\(\displaystyle \frac{2x}{3y} - \frac{x^2}{4y^3} + \frac{3}{5y^4} \)This is what I did, which is wrong according to the textbook. Could someone point out what I did wrong and how to correct it? Thanks.

\(\displaystyle \frac{(2x)(4y^3)-(x^2)(3y)}{(3y)(4y^3)} + \frac{3}{5y^4}\)

\(\displaystyle \frac{8xy^3-3x^2y}{(12y^3)} + \frac{3}{5y^4}\)

\(\displaystyle \frac{(8xy^3-3x^2y)(5y^4)+(3)(12y^4)}{(12y^3)(5y^4)}\)

\(\displaystyle \frac{(40xy^7-15x^2y^5)+36y^4}{60y^8}\)
Little things always mess you up.

Line 3, last term in the numerator. Check your power of y.

Last line, Check the power of y in the denominator.

Finally, there is some cancellation you can do.

-Dan
 
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  • #3
I would first observe that the lowest common denominator is $60y^4$ and to we may write the expression as:

\(\displaystyle \frac{2x}{3y}\cdot\frac{20y^3}{20y^3}-\frac{x^2}{4y^3}\cdot\frac{15y}{15y}+\frac{3}{5y^4}\cdot\frac{12}{12}\)

This is somewhat simpler than your method.

And so combining terms, what do we get?

As Dan stated, your expression is almost equivalent to this, you just need to divide each term in the numerator and denominator a common factor (after making the check Dan suggests).
 

FAQ: Simplify (Adding and Subtracting Rational Functions)

What are rational functions?

Rational functions are functions that can be written as a ratio of two polynomials. They typically have the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.

How do you simplify rational functions?

To simplify rational functions, we follow the steps of finding the common denominator, combining like terms, and factoring both the numerator and denominator. Then, we cancel out any common factors to simplify the expression.

Can you add or subtract rational functions?

Yes, we can add or subtract rational functions by following the same steps as simplifying them. We just need to make sure that we have a common denominator before adding or subtracting the fractions.

What should I do if the denominators of the rational functions are not the same?

If the denominators are not the same, we need to find the least common multiple (LCM) of the denominators and use it as the common denominator for both fractions. Then, we can proceed with adding or subtracting the fractions as usual.

Can rational functions ever be simplified to a single number?

Yes, in some cases, rational functions can be simplified to a single number. This usually happens when the numerator and denominator have the same degree, and the leading coefficients are not equal. In this case, the simplified rational function will have a horizontal asymptote at the ratio of the leading coefficients.

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