Find 'a' if the two polynomials `ax^(3)+3x^(2)-9and2x^(3)+4x+a`, leaves the same remainder when divided by x + 3.

Answer

To find the value of \( a \) such that the two polynomials \( f(x) = ax^3 + 3x^2 - 9 \) and \( g(x) = 2x^3 + 4x + a \) leave the same remainder when divided by \( x + 3 \), we will follow these steps:

### Step 1: Identify the Polynomials
Let:
- \( f(x) = ax^3 + 3x^2 - 9 \)
- \( g(x) = 2x^3 + 4x + a \)

### Step 2: Find the Value of \( x \)
Since we are dividing by \( x + 3 \), we set:
\[ x + 3 = 0 \]
Thus,
\[ x = -3 \]

### Step 3: Set the Remainders Equal
According to the problem, the remainders when \( f(x) \) and \( g(x) \) are evaluated at \( x = -3 \) must be equal:
\[ f(-3) = g(-3) \]

### Step 4: Calculate \( f(-3) \)
Substituting \( x = -3 \) into \( f(x) \):
\[ f(-3) = a(-3)^3 + 3(-3)^2 - 9 \]
Calculating each term:
\[ = a(-27) + 3(9) - 9 \]
\[ = -27a + 27 - 9 \]
\[ = -27a + 18 \]

### Step 5: Calculate \( g(-3) \)
Now substituting \( x = -3 \) into \( g(x) \):
\[ g(-3) = 2(-3)^3 + 4(-3) + a \]
Calculating each term:
\[ = 2(-27) + 4(-3) + a \]
\[ = -54 - 12 + a \]
\[ = -66 + a \]

### Step 6: Set the Equations Equal
Now we set the two results equal to each other:
\[ -27a + 18 = -66 + a \]

### Step 7: Solve for \( a \)
Rearranging the equation:
\[ -27a - a = -66 - 18 \]
\[ -28a = -84 \]
Dividing both sides by -28:
\[ a = \frac{-84}{-28} = 3 \]

Thus, the value of \( a \) is \( 3 \).

### Final Answer
The value of \( a \) is \( 3 \).

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