Show that (x-5) is a factor of `x^(3)-3x^(2)-4x-30`.

Answer

To show that \( (x - 5) \) is a factor of the polynomial \( x^3 - 3x^2 - 4x - 30 \), we can use the Factor Theorem. According to the Factor Theorem, \( (x - a) \) is a factor of a polynomial \( f(x) \) if and only if \( f(a) = 0 \).

### Step-by-Step Solution:

1. **Identify the Polynomial**:
Let \( f(x) = x^3 - 3x^2 - 4x - 30 \).

2. **Set Up for the Factor Theorem**:
We need to check if \( (x - 5) \) is a factor. According to the Factor Theorem, we need to evaluate \( f(5) \).

3. **Substitute \( x = 5 \) into the Polynomial**:
\[ f(5) = 5^3 - 3(5^2) - 4(5) - 30 \]

4. **Calculate Each Term**:
- Calculate \( 5^3 \):
\[ 5^3 = 125 \]
- Calculate \( 3(5^2) \):
\[ 5^2 = 25 \quad \Rightarrow \quad 3 \times 25 = 75 \]
- Calculate \( 4(5) \):
\[ 4 \times 5 = 20 \]

5. **Combine the Results**:
Now substitute these values back into the equation:
\[ f(5) = 125 - 75 - 20 - 30 \]

6. **Perform the Arithmetic**:
- First, combine the negative terms:
\[ -75 - 20 - 30 = -125 \]
- Now combine with \( 125 \):
\[ 125 - 125 = 0 \]

7. **Conclusion**:
Since \( f(5) = 0 \), by the Factor Theorem, \( (x - 5) \) is indeed a factor of the polynomial \( x^3 - 3x^2 - 4x - 30 \).

### Final Statement:
Thus, we have shown that \( (x - 5) \) is a factor of the polynomial \( x^3 - 3x^2 - 4x - 30 \).

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