Binomial Theorem
Expansion in powers of (1+x)
Grade 11
Question:
<p>If \(1 + x^4 + x^5 = a_0 + a_1(1+x) + a_2(1+x)^2 + a_3(1+x)^3 + a_4(1+x)^4 + a_5(1+x)^5\), then \(a_2\) equals:</p>
<p>4</p>
<p>\(-4\)</p>
<p>6</p>
<p>\(-6\)</p>
Step-by-Step Solution
Key Concept: Substitute y = 1+x (so x = y-1) into the left side and expand using binomial theorem to find coefficients of powers of y. The coefficient of y² gives a₂.
<p><strong>Step 1:</strong> Let y = 1+x, so x = y-1. Substitute into LHS:</p><p>1 + (y-1)⁴ + (y-1)⁵ = a₀ + a₁y + a₂y² + a₃y³ + a₄y⁴ + a₅y⁵</p><p><strong>Step 2:</strong> Expand (y-1)⁴ using binomial theorem:</p><p>(y-1)⁴ = y⁴ - 4y³ + 6y² - 4y + 1</p><p><strong>Step 3:</strong> Expand (y-1)⁵:</p><p>(y-1)⁵ = y⁵ - 5y⁴ + 10y³ - 10y² + 5y - 1</p><p><strong>Step 4:</strong> Add all three expressions:</p><p>LHS = 1 + (y⁴ - 4y³ + 6y² - 4y + 1) + (y⁵ - 5y⁴ + 10y³ - 10y² + 5y - 1)</p><p>= y⁵ + (1-5)y⁴ + (-4+10)y³ + (6-10)y² + (-4+5)y + (1+1-1)</p><p>= y⁵ - 4y⁴ + 6y³ - 4y² + y + 1</p><p><strong>Step 5:</strong> The coefficient of y² is a₂ = <strong>-4</strong></p><p>∴ Answer: B</p>
Correct Answer: B