Generalize the pattern for the division of (x^n)-(y^n) by x-y
set up a long algebraic divsion , hard to show in this format
I hope you know how to perform that
I got
x^n - y^n = (x-y)(x^(n-1) + yx^(n-2) + y^2 x^(n-3) + ... + y^(n-1) )
To generalize the pattern, let's consider the division of (x^n) - (y^n) by (x - y). We can apply the algebraic technique known as polynomial long division to find the quotient.
1. Start by writing the polynomial (x^n) - (y^n) as the dividend and (x - y) as the divisor.
2. Begin the long division process by dividing the first term of the dividend (x^n) by the first term of the divisor (x). The result is x^(n-1).
3. Multiply the divisor (x - y) by the result obtained in step 2, which is x^(n-1). This multiplication yields x^n - xy^(n-1).
4. Subtract the product obtained in step 3 from the dividend (x^n) - (y^n) to eliminate the first term in the dividend. The subtraction gives:
(x^n) - (y^n) - (x^n - xy^(n-1)) = xy^(n-1) - (y^n)
5. Bring down the next term from the dividend, which is y^n, and add it to the result obtained in step 4. The sum is xy^(n-1) + y^n.
6. Divide the first term of the new expression, xy^(n-1) + y^n, by the first term of the divisor (x). The result is y^(n-1).
7. Multiply the divisor (x - y) by the result obtained in step 6, which is y^(n-1). This multiplication yields xy^(n-1) - y^n.
8. Subtract the product obtained in step 7 from the new expression, xy^(n-1) + y^n, to eliminate the first term in the expression. The subtraction gives:
xy^(n-1) + y^n - (xy^(n-1) - y^n) = 2y^n
9. Since there are no more terms to bring down, the long division process is complete.
10. The quotient obtained from the long division is x^(n-1) + y^(n-1), and the remainder is 2y^n.
In summary, the division of (x^n) - (y^n) by (x - y) results in a quotient of x^(n-1) + y^(n-1) and a remainder of 2y^n.