<p>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:</p>
Step-by-Step Solution
Key Concept: Extract coefficients of x² and x³ from the product by identifying contributions from both factors: the polynomial (1 + ax + bx²) and the binomial expansion (1 - 3x)¹⁵. Since both coefficients must equal zero, set up two simultaneous equations.
<p><strong>Step 1:</strong> Find coefficients in (1 - 3x)¹⁵ using binomial theorem.</p><p>Coefficient of x¹: C(15,1)(-3)¹ = 15·(-3) = -45</p><p>Coefficient of x²: C(15,2)(-3)² = 105·9 = 945</p><p>Coefficient of x³: C(15,3)(-3)³ = 455·(-27) = -12285</p><p><strong>Step 2:</strong> In the product (1 + ax + bx²)(1 - 3x)¹⁵, find coefficient of x².</p><p>Coeff of x² = 1·(945) + a·(-45) + b·(1) = 945 - 45a + b = 0</p><p>∴ b - 45a = -945 ... (1)</p><p><strong>Step 3:</strong> Find coefficient of x³.</p><p>Coeff of x³ = 1·(-12285) + a·(945) + b·(-45) = -12285 + 945a - 45b = 0</p><p>∴ 945a - 45b = 12285 ... (2)</p><p><strong>Step 4:</strong> Solve the system. From equation (1): b = 45a - 945</p><p>Substitute into (2): 945a - 45(45a - 945) = 12285</p><p>945a - 2025a + 42525 = 12285</p><p>-1080a = -30240</p><p>a = 28</p><p><strong>Step 5:</strong> Find b: b = 45(28) - 945 = 1260 - 945 = 315</p><p>∴ Answer: (a, b) = (28, 315)</p>
Correct Answer: C