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

3 min read Jun 17, 2024

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

This article explores the process of dividing the polynomial (x^3 - 3x^2 - 5x - 25) by the binomial (x - 5). We'll use polynomial long division to achieve this.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division we learned in arithmetic. It involves the following steps:

Set up: Write the dividend (the polynomial being divided) inside the division symbol and the divisor (the polynomial dividing) outside.
Divide: Divide the leading term of the dividend by the leading term of the divisor. Write the quotient above the dividend.
Multiply: Multiply the quotient by the entire divisor. Write the product below the dividend.
Subtract: Subtract the product from the dividend.
Bring down: Bring down the next term of the dividend.
Repeat: Repeat steps 2-5 until there are no more terms to bring down.

Applying the Steps

Let's apply these steps to our problem:

          x^2 + 2x + 5 
      ____________________
x - 5 | x^3 - 3x^2 - 5x - 25
        -(x^3 - 5x^2)
        ____________________
              2x^2 - 5x 
              -(2x^2 - 10x)
              ____________________
                     5x - 25
                     -(5x - 25)
                     ____________________
                            0

Interpreting the Results

The result of our division is x^2 + 2x + 5. This means that:

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

We can also express this as:

(x^3 - 3x^2 - 5x - 25) = (x - 5)(x^2 + 2x + 5)

This tells us that (x - 5) is a factor of the polynomial (x^3 - 3x^2 - 5x - 25).

Conclusion

By using polynomial long division, we successfully divided the polynomial (x^3 - 3x^2 - 5x - 25) by the binomial (x - 5) and determined the quotient to be x^2 + 2x + 5. This process allows us to factorize polynomials and gain a deeper understanding of their relationships.