Polynomial Division: Simplifying 3rd Roots in Denominator

In summary, the expression with 3rd roots in the denominator can be rewritten as (x^2 + 5x + 25)/(x - 5) after using the identity y^3-a^3=(y-a)(y^2+ay+a^2) to simplify it.
  • #1
Hannisch
116
0
3rd roots in denominator

Homework Statement


[tex]\frac{ \sqrt[3]{25} + \sqrt[3]{5x} + \sqrt[3]{x^2} }{ \sqrt[3]{x} - \sqrt[3]{5} }[/tex]

Rewrite the expression with no roots in the denominator and it being simplified as far as possible.

Homework Equations


The Attempt at a Solution


I'm.. stumped. Seriously. How on Earth do I get rid of them? I can't do the conjugate, I can't take the denominator to the power of three, I can't take it times (x3 + 53) and I'm out of ideas.Sorry about it being called Polynomial division, mishap on my behalf.
 
Last edited:
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  • #2
Try to use the identity

y3-a3=(y-a)(y2+ay+a2)

ehild
 
  • #3
Thank you! I'd never have figured that one out myself
 

Related to Polynomial Division: Simplifying 3rd Roots in Denominator

1. What is polynomial division?

Polynomial division is the process of dividing one polynomial expression by another, resulting in a simplified polynomial expression.

2. What are 3rd roots in the denominator?

3rd roots in the denominator refer to the presence of a cubic root (or cube root) in the denominator of a fraction. In other words, the denominator contains a term that is raised to the power of 1/3.

3. How do you simplify 3rd roots in the denominator?

To simplify 3rd roots in the denominator, we use the concept of rationalizing the denominator. This involves multiplying both the numerator and denominator by the cube root of the number in the denominator to eliminate the 3rd root.

4. Can you give an example of simplifying 3rd roots in the denominator?

Yes, for example, if we have the fraction 1/(3√2), we can simplify it by multiplying both the numerator and denominator by √2, resulting in √2/(3√4). Then, we can simplify further by dividing √4 by 2, resulting in √2/6.

5. Why is it important to simplify 3rd roots in the denominator?

Simplifying 3rd roots in the denominator allows us to express fractions in their simplest form, making them easier to work with in mathematical calculations. It also helps us to avoid dealing with complex and unwieldy expressions.

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