Math Problem Statement

׳-1/125

Solution

The expression x31125x^3 - \frac{1}{125} can be factored using the difference of cubes formula. The difference of cubes formula is:

a3b3=(ab)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

In this case, we can rewrite 1125\frac{1}{125} as (15)3\left(\frac{1}{5}\right)^3. Thus, the expression becomes:

x3(15)3x^3 - \left(\frac{1}{5}\right)^3

Using the difference of cubes formula:

x3(15)3=(x15)(x2+x(15)+(15)2)x^3 - \left(\frac{1}{5}\right)^3 = \left( x - \frac{1}{5} \right)\left( x^2 + x\left(\frac{1}{5}\right) + \left(\frac{1}{5}\right)^2 \right)

Simplifying further:

x31125=(x15)(x2+15x+125)x^3 - \frac{1}{125} = \left( x - \frac{1}{5} \right)\left( x^2 + \frac{1}{5}x + \frac{1}{25} \right)

This is the factorized form of the expression x31125x^3 - \frac{1}{125}.

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:

  1. How do you factor expressions using the sum of cubes formula?
  2. What are some other examples of the difference of cubes?
  3. How can you use the difference of squares to factor expressions?
  4. What is the process for simplifying expressions with cube roots?
  5. How do you solve equations involving cube roots?
  6. Can the difference of cubes formula be used in solving cubic equations?
  7. How does the difference of cubes formula apply to polynomial division?
  8. What is the general method to factor a cubic polynomial?

Tip: Remember, the difference of cubes formula is a powerful tool for simplifying and solving cubic expressions. It's worth mastering!

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Cube Roots

Formulas

Difference of cubes formula

Theorems

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Suitable Grade Level

Grades 10-12