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(x^3-5x^2-7x+25)/(x-5)

4 min read Jun 17, 2024
(x^3-5x^2-7x+25)/(x-5)

Dividing Polynomials: (x^3-5x^2-7x+25)/(x-5)

This article will guide you through the process of dividing the polynomial (x^3-5x^2-7x+25) by (x-5) 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:

  1. Set up the division:

         ________
x - 5 | x^3 - 5x^2 - 7x + 25

  2. 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

  3. Multiply the divisor by the result:

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

  4. 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

  5. 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

  6. Repeat steps 2-5:

    • 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 (x-5) by the result (-7): (x-5)(-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

  7. The remainder:

    • The final result is -10, which is the remainder.

Final Result

Therefore, the result of dividing (x^3-5x^2-7x+25) by (x-5) is:

x^2 - 7 with a remainder of -10

This can also be written as:

(x^3-5x^2-7x+25)/(x-5) = x^2 - 7 - 10/(x-5)

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.

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