(x^2-10x+30)/(x-5)

3 min read Jun 17, 2024

Analyzing the Rational Expression (x^2 - 10x + 30) / (x - 5)

This article will delve into the analysis of the rational expression (x^2 - 10x + 30) / (x - 5). We will explore its domain, simplification, and potential for factorization.

Domain

The domain of a rational expression is the set of all possible values of x for which the expression is defined. In this case, the expression is undefined when the denominator (x - 5) equals zero. Therefore, the domain is all real numbers except for x = 5.

Simplification

The expression cannot be simplified further by factoring. This is because the quadratic expression in the numerator, x^2 - 10x + 30, does not factor into real numbers.

Polynomial Long Division

We can use polynomial long division to rewrite the expression. Here's how it works:

Set up the division:
```
    x - 3 
  x - 5 | x^2 - 10x + 30 
```

Divide the leading terms: x^2 divided by x is x.

    x - 3 
  x - 5 | x^2 - 10x + 30 
        -(x^2 - 5x)
        --------
               -5x + 30

Multiply the quotient (x) by the divisor (x - 5): x(x - 5) = x^2 - 5x
Subtract the result from the dividend: (x^2 - 10x + 30) - (x^2 - 5x) = -5x + 30
Bring down the next term (30): -5x + 30

Repeat steps 2-5: -5x divided by x is -5.

    x - 3 
  x - 5 | x^2 - 10x + 30 
        -(x^2 - 5x)
        --------
               -5x + 30
               -(-5x + 25)
               ---------
                       5

The remainder is 5: The division is complete.

This gives us the following result: (x^2 - 10x + 30) / (x - 5) = x - 3 + 5/(x - 5)

Conclusion

The expression (x^2 - 10x + 30) / (x - 5) is not factorable. We can rewrite it using polynomial long division as x - 3 + 5/(x - 5). Remember that the expression is undefined for x = 5.