If the polynomial `f (x) = x ^(3) + ax ^(2) + bx + 6` is divided by `(x - 3),` the remainder is 3. Also `(x - 2)` is a factor of the polynomial f (x). Find the vaue of a and b.

Options

A. `a =-1, b = 3`
B. `a =1, b =-3`
C. `a = 3, b =-1`
D. `a -3, b =-1`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) for the polynomial \( f(x) = x^3 + ax^2 + bx + 6 \) given the conditions that when divided by \( (x - 3) \), the remainder is 3, and \( (x - 2) \) is a factor of the polynomial.

### Step 1: Use the Remainder Theorem for \( (x - 3) \)

According to the Remainder Theorem, if a polynomial \( f(x) \) is divided by \( (x - c) \), the remainder is \( f(c) \). Here, we have:

\[ f(3) = 3 \]

Substituting \( x = 3 \) into the polynomial:

\[ f(3) = 3^3 + a(3^2) + b(3) + 6 \]
\[ = 27 + 9a + 3b + 6 \]
\[ = 33 + 9a + 3b \]

Setting this equal to the remainder:

\[ 33 + 9a + 3b = 3 \]

### Step 2: Simplify the Equation

Rearranging the equation gives:

\[ 9a + 3b = 3 - 33 \]
\[ 9a + 3b = -30 \]

Dividing the entire equation by 3:

\[ 3a + b = -10 \quad \text{(Equation 1)} \]

### Step 3: Use the Factor Condition for \( (x - 2) \)

Since \( (x - 2) \) is a factor of \( f(x) \), we have:

\[ f(2) = 0 \]

Substituting \( x = 2 \) into the polynomial:

\[ f(2) = 2^3 + a(2^2) + b(2) + 6 \]
\[ = 8 + 4a + 2b + 6 \]
\[ = 14 + 4a + 2b \]

Setting this equal to zero:

\[ 14 + 4a + 2b = 0 \]

### Step 4: Simplify the Second Equation

Rearranging gives:

\[ 4a + 2b = -14 \]

Dividing the entire equation by 2:

\[ 2a + b = -7 \quad \text{(Equation 2)} \]

### Step 5: Solve the System of Equations

Now we have two equations:

1. \( 3a + b = -10 \) (Equation 1)
2. \( 2a + b = -7 \) (Equation 2)

We can eliminate \( b \) by subtracting Equation 2 from Equation 1:

\[ (3a + b) - (2a + b) = -10 - (-7) \]
\[ 3a - 2a = -10 + 7 \]
\[ a = -3 \]

### Step 6: Substitute \( a \) Back to Find \( b \)

Now substitute \( a = -3 \) back into Equation 1:

\[ 3(-3) + b = -10 \]
\[ -9 + b = -10 \]
\[ b = -10 + 9 \]
\[ b = -1 \]

### Final Answer

Thus, the values of \( a \) and \( b \) are:

\[ a = -3, \quad b = -1 \]