Find the value of 'm'. If `mx^3+ 2x^2- 3 and x^2- mx+ 4` leave the same remainder when each is divided by x- 2.

Answer

To find the value of 'm' such that the polynomials \( mx^3 + 2x^2 - 3 \) and \( x^2 - mx + 4 \) leave the same remainder when divided by \( x - 2 \), we can use the Remainder Theorem. According to the theorem, the remainder of a polynomial \( f(x) \) when divided by \( x - a \) is \( f(a) \).

### Step-by-step Solution:

1. **Identify the Polynomials**:
We have two polynomials:
- \( f(x) = mx^3 + 2x^2 - 3 \)
- \( g(x) = x^2 - mx + 4 \)

2. **Apply the Remainder Theorem**:
We need to find the remainders of both polynomials when evaluated at \( x = 2 \).

3. **Calculate the Remainder of \( f(x) \)**:
Substitute \( x = 2 \) into \( f(x) \):
\[ f(2) = m(2)^3 + 2(2)^2 - 3 \]
\[ = m(8) + 2(4) - 3 \]
\[ = 8m + 8 - 3 \]
\[ = 8m + 5 \]

4. **Calculate the Remainder of \( g(x) \)**:
Substitute \( x = 2 \) into \( g(x) \):
\[ g(2) = (2)^2 - m(2) + 4 \]
\[ = 4 - 2m + 4 \]
\[ = 8 - 2m \]

5. **Set the Remainders Equal**:
Since both polynomials leave the same remainder when divided by \( x - 2 \), we set the two remainders equal to each other:
\[ 8m + 5 = 8 - 2m \]

6. **Solve for \( m \)**:
Rearranging the equation:
\[ 8m + 2m = 8 - 5 \]
\[ 10m = 3 \]
\[ m = \frac{3}{10} \]

### Final Answer:
The value of \( m \) is \( \frac{3}{10} \).

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