How Many Values Of $x$ Must Be Excluded In The Expression $\frac{x-2}{(x+9)(x-5)}$?A. 0 B. 1 C. 2 D. 3

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Understanding the Problem

When dealing with rational expressions, it's essential to consider the values of the variable that would make the denominator equal to zero. These values are called excluded values or restricted values, and they must be excluded from the domain of the expression.

The Expression and Its Denominator

The given expression is $\frac{x-2}{(x+9)(x-5)}$. To find the excluded values, we need to focus on the denominator, which is $(x+9)(x-5)$. We want to find the values of xx that would make this expression equal to zero.

Finding the Excluded Values

To find the excluded values, we need to set the denominator equal to zero and solve for xx. We can do this by setting each factor equal to zero and solving for xx.

Solving for x in the First Factor

The first factor is x+9x+9. To find the value of xx that would make this factor equal to zero, we can set it equal to zero and solve for xx:

x+9=0x+9=0

Subtracting 9 from both sides gives us:

x=−9x=-9

So, the value of xx that would make the first factor equal to zero is x=−9x=-9.

Solving for x in the Second Factor

The second factor is x−5x-5. To find the value of xx that would make this factor equal to zero, we can set it equal to zero and solve for xx:

x−5=0x-5=0

Adding 5 to both sides gives us:

x=5x=5

So, the value of xx that would make the second factor equal to zero is x=5x=5.

Conclusion

We have found two values of xx that would make the denominator equal to zero: x=−9x=-9 and x=5x=5. These values must be excluded from the domain of the expression.

The Final Answer

Based on our analysis, we can conclude that there are 2 values of xx that must be excluded in the expression.

Comparison with the Options

Let's compare our answer with the options provided:

  • A. 0: This is incorrect, as we have found two excluded values.
  • B. 1: This is incorrect, as we have found two excluded values.
  • C. 2: This is correct, as we have found two excluded values.
  • D. 3: This is incorrect, as we have found only two excluded values.

The final answer is C.