Read the statements carefully and select the correct option.
Statement-I : If `p(x) and g(x)` are two polynomials such that degree of `p(x) ge`. degree of `g(x) and g(x) ne 0`, then we can find polynomials `q(x) and r(x)` such that `p(x) = g(x) · q(x) + r(x)` , where `r(x) = 0` or degree of `r(x) =0` degree of `g(x)`.
Statement-II : `3x^(2) + x-1 = ( x + 1) ( 3x -2) + 1`.

Options

A. Both Statement-I and Statement-II are true.
B. Both Statement-I and Statement-II are false.
C. Statement-I is true but Statement-II is false.
D. Statement-I is false but Statement-II is true.

Answer

To solve the problem, we need to analyze both statements carefully.

### Step 1: Understanding Statement-I
Statement-I states that if \( p(x) \) and \( g(x) \) are two polynomials such that the degree of \( p(x) \) is greater than or equal to the degree of \( g(x) \) and \( g(x) \neq 0 \), then we can find polynomials \( q(x) \) and \( r(x) \) such that:
\[ p(x) = g(x) \cdot q(x) + r(x) \]
where \( r(x) = 0 \) or the degree of \( r(x) \) is less than the degree of \( g(x) \).

### Step 2: Analyzing the Polynomial Division
This is essentially a statement about polynomial long division. The polynomial division theorem states that for any two polynomials \( p(x) \) and \( g(x) \) (where \( g(x) \neq 0 \)), there exist unique polynomials \( q(x) \) (the quotient) and \( r(x) \) (the remainder) such that:
- The degree of \( r(x) \) is less than the degree of \( g(x) \).
- If \( r(x) = 0 \), it indicates that \( g(x) \) divides \( p(x) \) completely.

### Step 3: Understanding Statement-II
Statement-II gives an example:
\[ 3x^2 + x - 1 = (x + 1)(3x - 2) + 1 \]
Here, we can observe that:
- The polynomial \( 3x^2 + x - 1 \) is being expressed in terms of \( (x + 1) \) and \( (3x - 2) \).
- The remainder \( r(x) \) is \( 1 \), which is of degree \( 0 \) (since it's a constant).

### Step 4: Conclusion
Both statements are consistent with the polynomial division theorem. Therefore, we conclude that:

- **Statement-I is true** because it accurately describes the polynomial division process.
- **Statement-II is also true** as it provides a specific example that adheres to the rules outlined in Statement-I.

Thus, both statements are correct.

### Final Answer
Both Statement-I and Statement-II are true.

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