Key Concept: Binomial Theorem and Coefficient Extraction

This problem requires the application of the Binomial Theorem to expand a binomial expression and then careful extraction of specific coefficients from the product of two polynomials. The general term in the binomial expansion of $(X+Y)^n$ is given by $T_{r+1} = \binom{n}{r} X^{n-r} Y^r$. When dealing with a product of polynomials, the coefficient of a specific power of $x$ is found by summing the products of terms from each polynomial that multiply to give that power of $x$.

For the expression $(1 - 3x)^{15}$, we can use the Binomial Theorem where $X=1$, $Y=-3x$, and $n=15$. The general term, $T_{r+1}$, is: $T_{r+1} = \binom{15}{r} (1)^{15-r} (-3x)^r = \binom{15}{r} (-3)^r x^r$ This formula will be used to find the coefficients of $x^0$, $x^1$, $x^2$, and $x^3$ in the expansion of $(1 - 3x)^{15}$.

Step 1: Determine the Coefficients of Powers of $x$ from $(1 - 3x)^{15}$

Let's find the required coefficients from the binomial expansion of $(1 - 3x)^{15}$:

• Coefficient of $x^0$ (constant term): Set $r=0$. $\binom{15}{0} (-3)^0 x^0 = 1 \times 1 \times 1 = 1$
• Coefficient of $x^1$: Set $r=1$. $\binom{15}{1} (-3)^1 x^1 = 15 \times (-3) \times x^1 = -45x$ The coefficient of $x^1$ is $-45$.
• Coefficient of $x^2$: Set $r=2$. $\binom{15}{2} (-3)^2 x^2 = \frac{15 \times 14}{2} \times 9 \times x^2 = (15 \times 7) \times 9 \times x^2 = 105 \times 9 \times x^2 = 945x^2$ The coefficient of $x^2$ is $945$.
• Coefficient of $x^3$: Set $r=3$. $\binom{15}{3} (-3)^3 x^3 = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} \times (-27) \times x^3 = (5 \times 7 \times 13) \times (-27) \times x^3 = 455 \times (-27) \times x^3 = -12285x^3$ The coefficient of $x^3$ is $-12285$.

Step 2: Calculate the Coefficient of $x^2$ in the Full Expansion

The given expression is $(1 + ax + bx^2)(1 - 3x)^{15}$. To find the coefficient of $x^2$, we need to identify all combinations of terms from $(1 + ax + bx^2)$ and $(1 - 3x)^{15}$ that multiply to produce an $x^2$ term.

1. Multiply the constant term of $(1 + ax + bx^2)$, which is $1$, by the coefficient of $x^2$ in $(1 - 3x)^{15}$. Contribution: $1 \times 945 = 945$.
2. Multiply the $x$ term of $(1 + ax + bx^2)$, which is $ax$, by the coefficient of $x^1$ in $(1 - 3x)^{15}$. Contribution: $a \times (-45) = -45a$.
3. Multiply the $x^2$ term of $(1 + ax + bx^2)$, which is $bx^2$, by the coefficient of $x^0$ in $(1 - 3x)^{15}$. Contribution: $b \times 1 = b$.

Summing these contributions, the total coefficient of $x^2$ is: $945 - 45a + b$ Given that this coefficient is zero: $945 - 45a + b = 0 \quad \text{(Equation 1)}$

Step 3: Calculate the Coefficient of $x^3$ in the Full Expansion

Similarly, to find the coefficient of $x^3$, we identify all combinations of terms that multiply to produce an $x^3$ term:

1. Multiply the constant term of $(1 + ax + bx^2)$, which is $1$, by the coefficient of $x^3$ in $(1 - 3x)^{15}$. Contribution: $1 \times (-12285) = -12285$.
2. Multiply the $x$ term of $(1 + ax + bx^2)$, which is $ax$, by the coefficient of $x^2$ in $(1 - 3x)^{15}$. Contribution: $a \times 945 = 945a$.
3. Multiply the $x^2$ term of $(1 + ax + bx^2)$, which is $bx^2$, by the coefficient of $x^1$ in $(1 - 3x)^{15}$. Contribution: $b \times (-45) = -45b$.

Summing these contributions, the total coefficient of $x^3$ is: $-12285 + 945a - 45b$ Given that this coefficient is zero: $-12285 + 945a - 45b = 0$ To simplify, we can divide the entire equation by $45$: $\frac{-12285}{45} + \frac{945a}{45} - \frac{45b}{45} = 0$ $-273 + 21a - b = 0$ Rearranging this gives: $21a - b = 273 \quad \text{(Equation 2)}$

Step 4: Solve the System of Linear Equations

We now have a system of two linear equations with two variables, $a$ and $b$:

1. $945 - 45a + b = 0$
2. $21a - b = 273$

From Equation 1, we can express $b$ in terms of $a$: $b = 45a - 945$ Substitute this expression for $b$ into Equation 2: $21a - (45a - 945) = 273$ Distribute the negative sign: $21a - 45a + 945 = 273$ Combine like terms: $-24a + 945 = 273$ Subtract $945$ from both sides: $-24a = 273 - 945$ $-24a = -672$ Divide by $-24$ to find $a$: $a = \frac{-672}{-24} = 28$ Now substitute the value of $a=28$ back into the expression for $b$: $b = 45(28) - 945$ $b = 1260 - 945$ $b = 315$ Thus, the ordered pair $(a,b)$ is $(28, 315)$.

Summary and Key Takeaway

By systematically applying the Binomial Theorem to find coefficients of specific powers of $x$ in the expansion of $(1 - 3x)^{15}$, and then carefully combining terms from the product $(1 + ax + bx^2)(1 - 3x)^{15}$, we formed a system of linear equations. Solving this system yielded the values for $a$ and $b$. The critical steps involved precise calculation of binomial coefficients and powers, meticulous tracking of terms, and accurate algebraic manipulation to solve the simultaneous equations.

The ordered pair $(a,b)$ is $(28, 315)$, which corresponds to option (B).

Common Mistakes to Avoid:

• Sign Errors: Be very careful with negative signs, especially when terms like $(-3x)^r$ are involved.
• Calculation Errors: Double-check calculations for combinations ($\binom{n}{r}$) and multiplication.
• Missing Terms: Ensure all possible combinations of terms that contribute to the desired power of $x$ are included.
• Algebraic Errors: Take care when solving simultaneous equations, especially during substitution and simplification.