If the coefficients of `x^(2)` and `x^(3)`are both zero, in the expansion of the expression `(1 + ax + bx^(2)) (1 - 3x)^15` in powers of x, then the ordered pair (a,b) is equal to

Options

A. (a)`(-21, 714)`
B. (b)(-54, 315)
C. (c)(28, 861)
D. (d)(28, 315)

Explanation

To solve the problem, we need to find the ordered pair \((a, b)\) such that the coefficients of \(x^2\) and \(x^3\) in the expansion of \((1 + ax + bx^2)(1 - 3x)^{15}\) are both zero.

### Step 1: Find the coefficient of \(x^2\)

The coefficient of \(x^2\) in the expansion can be obtained from the following contributions:
1. From \(1\) in \((1 + ax + bx^2)\) and the \(x^2\) term from \((1 - 3x)^{15}\).
2. From \(ax\) in \((1 + ax + bx^2)\) and the \(x^1\) term from \((1 - 3x)^{15}\).
3. From \(bx^2\) in \((1 + ax + bx^2)\) and the constant term from \((1 - 3x)^{15}\).

The coefficient of \(x^2\) in \((1 - 3x)^{15}\) is given by:
\[ \binom{15}{2} (-3)^2 = 15 \cdot 6 = 90 \]
The coefficient of \(x^1\) is:
\[ \binom{15}{1} (-3) = -45 \]
The constant term is:
\[ \binom{15}{0} = 1 \]

Thus, the coefficient of \(x^2\) is:
\[ 90 + (-45)a + b = 0 \]
This simplifies to:
\[ -45a + b + 90 = 0 \quad \text{(Equation 1)} \]

### Step 2: Find the coefficient of \(x^3\)

The coefficient of \(x^3\) can be obtained from:
1. From \(1\) in \((1 + ax + bx^2)\) and the \(x^3\) term from \((1 - 3x)^{15}\).
2. From \(ax\) in \((1 + ax + bx^2)\) and the \(x^2\) term from \((1 - 3x)^{15}\).
3. From \(bx^2\) in \((1 + ax + bx^2)\) and the \(x^1\) term from \((1 - 3x)^{15}\).

The coefficient of \(x^3\) in \((1 - 3x)^{15}\) is:
\[ \binom{15}{3} (-3)^3 = 455 \cdot (-27) = -12285 \]
The coefficient of \(x^2\) is:
\[ 90 \quad \text{(from previous calculation)} \]
The coefficient of \(x^1\) is:
\[ -45 \]

Thus, the coefficient of \(x^3\) is:
\[ -12285 + (-45)a + (-3b) = 0 \]
This simplifies to:
\[ -45a - 3b - 12285 = 0 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations

Now we have the two equations:
1. \(-45a + b + 90 = 0\)
2. \(-45a - 3b - 12285 = 0\)

From Equation 1, we can express \(b\) in terms of \(a\):
\[ b = 45a - 90 \]

Substituting this into Equation 2:
\[ -45a - 3(45a - 90) - 12285 = 0 \]
Expanding this gives:
\[ -45a - 135a + 270 - 12285 = 0 \]
Combining like terms:
\[ -180a + 270 - 12285 = 0 \]
\[ -180a = 12285 - 270 \]
\[ -180a = 12015 \]
\[ a = -\frac{12015}{180} = -66.75 \]

Now substituting \(a\) back into the equation for \(b\):
\[ b = 45(-66.75) - 90 \]
Calculating gives:
\[ b = -3003 - 90 = -3093 \]

Thus, the ordered pair \((a, b)\) is:
\[ (-66.75, -3093) \]

### Final Answer
The ordered pair \((a, b)\) is:
\[ \boxed{(-66.75, -3093)} \]