When `x^(5)-5x^(4)+9x^(3)-6x^(2)-16x+13` is divided by `x^(2)-3x+a` , then quotient and remainders are `x^(3)-2x^(2)+x+1` and `-15x+11` respectively . Find the value of a .

Options

A. 2
B. 1
C. 3
D. 4

Explanation

To find the value of \( a \) in the polynomial division problem, we start with the given polynomial:

\[ P(x) = x^5 - 5x^4 + 9x^3 - 6x^2 - 16x + 13 \]

and we know that when \( P(x) \) is divided by \( D(x) = x^2 - 3x + a \), the quotient \( Q(x) \) and remainder \( R(x) \) are given as:

\[ Q(x) = x^3 - 2x^2 + x + 1 \]
\[ R(x) = -15x + 11 \]

According to the polynomial division theorem, we can express \( P(x) \) as:

\[ P(x) = Q(x) \cdot D(x) + R(x) \]

Substituting the known values into this equation gives us:

\[ x^5 - 5x^4 + 9x^3 - 6x^2 - 16x + 13 = (x^3 - 2x^2 + x + 1)(x^2 - 3x + a) + (-15x + 11) \]

Now, we will expand the right-hand side of the equation.

1. **Multiply \( Q(x) \) by \( D(x) \)**:

\[ (x^3 - 2x^2 + x + 1)(x^2 - 3x + a) \]

We will distribute each term in \( Q(x) \) with each term in \( D(x) \):

- \( x^3 \cdot (x^2 - 3x + a) = x^5 - 3x^4 + ax^3 \)
- \( -2x^2 \cdot (x^2 - 3x + a) = -2x^4 + 6x^3 - 2ax^2 \)
- \( x \cdot (x^2 - 3x + a) = x^3 - 3x^2 + ax \)
- \( 1 \cdot (x^2 - 3x + a) = x^2 - 3x + a \)

Now, combine all these results:

\[ x^5 + (-3x^4 - 2x^4) + (ax^3 + 6x^3 + x^3) + (-2ax^2 - 3x^2 + x^2) + (ax - 3x + a) \]

Simplifying this gives:

\[ x^5 - 5x^4 + (a + 7)x^3 + (-2a - 2)x^2 + (a - 3 - 15)x + a + 11 \]

2. **Combine with the remainder**:

Now, we add the remainder \( R(x) = -15x + 11 \):

\[ P(x) = x^5 - 5x^4 + (a + 7)x^3 + (-2a - 2)x^2 + (a - 18)x + (a + 22) \]

3. **Set coefficients equal**:

Now we will equate the coefficients of \( P(x) \) with the expanded polynomial:

- Coefficient of \( x^3 \):
\[ a + 7 = 9 \implies a = 2 \]

- Coefficient of \( x^2 \):
\[ -2a - 2 = -6 \implies -2(2) - 2 = -6 \quad \text{(which is consistent)} \]

- Coefficient of \( x \):
\[ a - 18 = -16 \implies 2 - 18 = -16 \quad \text{(which is consistent)} \]

- Constant term:
\[ a + 22 = 13 \implies 2 + 22 = 24 \quad \text{(not needed as we already found \( a \))} \]

Thus, the value of \( a \) is:

\[ \boxed{2} \]