Dividing Polynomials: (x^35x^27x+25)/(x5)
This article will guide you through the process of dividing the polynomial (x^35x^27x+25) by (x5) using polynomial long division.
Understanding Polynomial Long Division
Polynomial long division is a method for dividing polynomials, similar to the long division used for numbers. The process involves repeatedly dividing the leading term of the dividend by the leading term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.
Steps for Polynomial Long Division
Here's how to perform the division:

Set up the division:
________ x  5  x^3  5x^2  7x + 25

Divide the leading terms:
 Divide the leading term of the dividend (x^3) by the leading term of the divisor (x): x^3 / x = x^2.
 Write the result (x^2) above the x^2 term in the dividend.
x^2 x  5  x^3  5x^2  7x + 25

Multiply the divisor by the result:
 Multiply the divisor (x5) by the result (x^2): (x5)(x^2) = x^3  5x^2.
x^2 x  5  x^3  5x^2  7x + 25 x^3  5x^2

Subtract:
 Subtract the product (x^3  5x^2) from the dividend. Notice that the x^3 and 5x^2 terms cancel out.
x^2 x  5  x^3  5x^2  7x + 25 x^3  5x^2  7x + 25

Bring down the next term:
 Bring down the next term (7x) from the dividend.
x^2 x  5  x^3  5x^2  7x + 25 x^3  5x^2  7x + 25

Repeat steps 25:
 Divide the new leading term (7x) by the leading term of the divisor (x): 7x / x = 7.
 Write the result (7) above the x term in the dividend.
 Multiply the divisor (x5) by the result (7): (x5)(7) = 7x + 35
 Subtract the product (7x + 35) from the current dividend.
x^2  7 x  5  x^3  5x^2  7x + 25 x^3  5x^2  7x + 25 7x + 35  10

The remainder:
 The final result is 10, which is the remainder.
Final Result
Therefore, the result of dividing (x^35x^27x+25) by (x5) is:
x^2  7 with a remainder of 10
This can also be written as:
(x^35x^27x+25)/(x5) = x^2  7  10/(x5)
Conclusion
Polynomial long division is a valuable tool for simplifying polynomial expressions and understanding their relationships. By following the steps outlined above, you can successfully divide any polynomial by another polynomial.