When the polynomial `x^(3)+2x^(2)-5ax-7` is divided by (x - 1), the remainder is A and when the polynomial. `x^3+ax^2-12x+16` is divided by (x + 2), the remainder is B. Find the value of 'a' if 2A + B = 0.

Answer

To solve the problem, we will use the Remainder Theorem, which states that the remainder of a polynomial \( P(x) \) when divided by \( x - c \) is \( P(c) \).

### Step 1: Find the remainder \( A \) when \( P_1(x) = x^3 + 2x^2 - 5ax - 7 \) is divided by \( x - 1 \)

Using the Remainder Theorem:
\[ A = P_1(1) = 1^3 + 2(1^2) - 5a(1) - 7 \]
Calculating this:
\[ A = 1 + 2 - 5a - 7 = -5a - 4 \]

### Step 2: Find the remainder \( B \) when \( P_2(x) = x^3 + ax^2 - 12x + 16 \) is divided by \( x + 2 \)

Again, using the Remainder Theorem:
\[ B = P_2(-2) = (-2)^3 + a(-2)^2 - 12(-2) + 16 \]
Calculating this:
\[ B = -8 + 4a + 24 + 16 = 4a + 32 \]

### Step 3: Set up the equation from the given condition \( 2A + B = 0 \)

Substituting the values of \( A \) and \( B \):
\[ 2(-5a - 4) + (4a + 32) = 0 \]
Expanding this:
\[ -10a - 8 + 4a + 32 = 0 \]
Combining like terms:
\[ -6a + 24 = 0 \]

### Step 4: Solve for \( a \)

Rearranging gives:
\[ -6a = -24 \implies a = 4 \]

### Final Answer
The value of \( a \) is \( 4 \).

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