(x%5E4%2Bx%5E3y%2Bx%5E2y%5E2%2Bxy%5E3%2By%5E4)%3Dx%5E5-y%5E5.jpeg "(x-y)(x^4+x^3y+x^2y^2+xy^3+y^4)=x^5-y^5")
Understanding the Difference of Fifth Powers: (x-y)(x^4+x^3y+x^2y^2+xy^3+y^4)=x^5-y^5
This equation represents a fundamental pattern in algebra known as the difference of fifth powers. It demonstrates how to factor the difference of two perfect fifth powers.
The Pattern
The equation itself highlights a key pattern:
(x-y)(x^4+x^3y+x^2y^2+xy^3+y^4) = x^5 - y^5
Explanation:
- Left-hand side:
- The first factor (x-y) is the difference of two terms.
- The second factor is a polynomial with terms that follow a specific structure:
- Each term contains powers of x and y, where the exponents of x decrease by one and the exponents of y increase by one.
- The coefficients are all 1.
- Right-hand side:
- The right-hand side is simply the difference of two perfect fifth powers: x^5 and y^5.
The Significance of the Pattern
This equation is a powerful tool for:
- Factoring expressions: It allows you to quickly factor the difference of two fifth powers into a binomial and a polynomial.
- Simplifying expressions: You can use this equation to simplify expressions involving the difference of fifth powers.
- Solving equations: By factoring expressions, you can often solve equations more easily.
Example
Let's see how this equation can be used to factor the expression:
x^5 - 32
We recognize that 32 is the fifth power of 2 (2^5). Applying the pattern, we can factor the expression as follows:
x^5 - 32 = (x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)
Conclusion
The equation (x-y)(x^4+x^3y+x^2y^2+xy^3+y^4)=x^5-y^5 provides a valuable shortcut for factoring and simplifying expressions involving the difference of fifth powers. It is a fundamental pattern that helps build a deeper understanding of algebraic manipulations and problem-solving strategies.