find the value of k so that when (3x^2+kx+5)/(x-1) the remainder is equal to 2

To find the value of \( k \) such that the remainder of the division of the polynomial \( \frac{3x^2 + kx + 5}{x - 1} \) is equal to 2, we can use the Remainder Theorem.

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder of that division is \( f(a) \). In this case, the polynomial is \( 3x^2 + kx + 5 \) and \( a = 1 \) (since we are dividing by \( x - 1 \)).

### Step-by-Step Solution

1. **Apply the Remainder Theorem**:  
   Substitute \( x = 1 \) into the polynomial:

   \[
   f(1) = 3(1)^2 + k(1) + 5
   \]

2. **Set up the equation with the remainder**:

   Since the remainder is 2:

   \[
   3(1)^2 + k(1) + 5 = 2
   \]

   Simplify the expression:

   \[
   3 + k + 5 = 2
   \]

   \[
   k + 8 = 2
   \]

3. **Solve for \( k \)**:

   \[
   k = 2 - 8
   \]

   \[
   k = -6
   \]

### Conclusion

The value of \( k \) that makes the remainder equal to 2 when dividing the polynomial \( 3x^2 + kx + 5 \) by \( x - 1 \) is \( \boxed{-6} \).

Would you like more details, or do you have any questions? Here are some related topics you might find interesting:

1. How to use the Remainder Theorem with different polynomials.
2. Understanding the Factor Theorem and its connection to the Remainder Theorem.
3. How to perform polynomial long division step-by-step.
4. Examples of finding unknown coefficients in polynomials.
5. Exploring synthetic division as an alternative method to find remainders.

**Tip:** When applying the Remainder Theorem, always substitute the value of \( x \) that makes the divisor zero!

Find the value of k so that when (3x^2 + kx + 5)/(x-1) the remainder is equal to 2.

Discover how to find the value of k in the polynomial 3x^2 + kx + 5, so that when divided by (x - 1), the remainder is 2. This problem involves applying the Remainder Theorem and algebraic techniques, suitable for Grades 9-10.

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