Using remainder theorem, find the value of k if on dividing `2x^3+ 3x^2-kx + 5` by x-2. leaves a remainder 7

Answer

To solve the problem using the Remainder Theorem, we follow these steps:

### Step 1: Identify the polynomial and the divisor
We are given the polynomial \( f(x) = 2x^3 + 3x^2 - kx + 5 \) and the divisor \( x - 2 \).

### Step 2: Apply the Remainder Theorem
According to the Remainder Theorem, when a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). Here, \( a = 2 \).

### Step 3: Substitute \( x = 2 \) into the polynomial
We need to find \( f(2) \):
\[ f(2) = 2(2)^3 + 3(2)^2 - k(2) + 5 \]
Calculating each term:
- \( 2(2)^3 = 2 \times 8 = 16 \)
- \( 3(2)^2 = 3 \times 4 = 12 \)
- \( -k(2) = -2k \)
- The constant term is \( 5 \)

So,
\[ f(2) = 16 + 12 - 2k + 5 \]

### Step 4: Combine the terms
Now combine the constant terms:
\[ f(2) = 33 - 2k \]

### Step 5: Set the remainder equal to 7
According to the problem, the remainder when dividing by \( x - 2 \) is 7:
\[ 33 - 2k = 7 \]

### Step 6: Solve for \( k \)
Now, we solve for \( k \):
\[ 33 - 2k = 7 \]
Subtract 33 from both sides:
\[ -2k = 7 - 33 \]
\[ -2k = -26 \]
Now, divide both sides by -2:
\[ k = \frac{-26}{-2} = 13 \]

### Final Answer
Thus, the value of \( k \) is \( 13 \).

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