In each of the following cases (Q.9-12), find whether g(x) is a factor of p(x) :
`p(x)=3x^(3)+5x^(2)-7x-1, " " g(x)=x-1`

Answer

To determine whether \( g(x) = x - 1 \) is a factor of \( p(x) = 3x^3 + 5x^2 - 7x - 1 \), we will use the Factor Theorem. According to the Factor Theorem, \( g(x) \) is a factor of \( p(x) \) if \( p(a) = 0 \), where \( a \) is the root of \( g(x) \).

### Step-by-step Solution:

**Step 1: Identify the root of \( g(x) \)**
Since \( g(x) = x - 1 \), we set \( g(x) = 0 \) to find the root:
\[ x - 1 = 0 \implies x = 1 \]

**Step 2: Substitute the root into \( p(x) \)**
Now, we need to evaluate \( p(1) \):
\[ p(x) = 3x^3 + 5x^2 - 7x - 1 \]
Substituting \( x = 1 \):
\[ p(1) = 3(1)^3 + 5(1)^2 - 7(1) - 1 \]

**Step 3: Calculate \( p(1) \)**
Now, we calculate each term:
\[ = 3(1) + 5(1) - 7 - 1 \]
\[ = 3 + 5 - 7 - 1 \]
\[ = 8 - 8 = 0 \]

**Step 4: Conclusion**
Since \( p(1) = 0 \), by the Factor Theorem, we conclude that \( g(x) = x - 1 \) is indeed a factor of \( p(x) \).

### Final Answer:
Thus, \( x - 1 \) is a factor of \( 3x^3 + 5x^2 - 7x - 1 \).

---