The polynomial `p(x)=x^(4)-2x^(3)+3x^(2)-ax+3a` when divided by `(x+1)` leaves the remainder `19`. Find the value of a .

Options

A. `19//4`
B. `13//4`
C. `1//4`
D. `17//4`

Answer

To solve the problem, we will use the Remainder Theorem, which states that the remainder of the division of a polynomial \( p(x) \) by \( x - c \) is equal to \( p(c) \).

Given the polynomial:
\[ p(x) = x^4 - 2x^3 + 3x^2 - ax + 3a \]
we need to find the value of \( a \) such that when \( p(x) \) is divided by \( x + 1 \), the remainder is 19.

### Step 1: Apply the Remainder Theorem
Since we are dividing by \( x + 1 \), we will evaluate \( p(-1) \):
\[ p(-1) = (-1)^4 - 2(-1)^3 + 3(-1)^2 - a(-1) + 3a \]

### Step 2: Calculate \( p(-1) \)
Now, we calculate each term:
- \( (-1)^4 = 1 \)
- \( -2(-1)^3 = -2(-1) = 2 \)
- \( 3(-1)^2 = 3(1) = 3 \)
- \( -a(-1) = a \)
- \( 3a \)

Putting it all together:
\[ p(-1) = 1 + 2 + 3 + a + 3a = 6 + 4a \]

### Step 3: Set the equation for the remainder
According to the problem, the remainder when \( p(x) \) is divided by \( x + 1 \) is 19. Therefore, we set up the equation:
\[ 6 + 4a = 19 \]

### Step 4: Solve for \( a \)
Now, we will solve for \( a \):
\[ 4a = 19 - 6 \]
\[ 4a = 13 \]
\[ a = \frac{13}{4} \]

### Conclusion
The value of \( a \) is \( \frac{13}{4} \).