Binomial Theorem
Coefficient in polynomial expansion
Grade 11
Question:
<p>Let \((2x^2 + 3x + 4)^{10} = \sum_{r=0}^{20} a_r x^r\) where \(a_0, a_1, a_2, \ldots, a_{20}\) are constant, then find the value of \(\dfrac{a_7}{a_{13}}\).</p>
Step-by-Step Solution
Key Concept: Use the property that coefficients in polynomial expansions satisfy a reciprocal relationship when the polynomial has a specific symmetry. For a polynomial P(x), the coefficient of x^r in [P(x)]^n relates to the coefficient of x^(kn-r) through substitution of x with 1/x.
<p><strong>Step 1: Identify the structure</strong></p><p>We have (2x² + 3x + 4)^10 = Σ a_r x^r where r goes from 0 to 20.</p><p>The maximum power is 20 since (x²)^10 = x^20, and the polynomial is degree 2, so the expansion has 2(10) + 1 = 21 terms.</p><p><strong>Step 2: Use the reciprocal property</strong></p><p>Consider that if (2x² + 3x + 4)^10 = Σ a_r x^r, then replacing x with 1/x:</p><p>(2/x² + 3/x + 4)^10 = Σ a_r/x^r</p><p>Multiplying both sides by x^20:</p><p>x^20(2/x² + 3/x + 4)^10 = Σ a_r x^(20-r)</p><p>(2 + 3x + 4x²)^10 = Σ a_r x^(20-r)</p><p><strong>Step 3: Relate the two expansions</strong></p><p>Notice that (4x² + 3x + 2)^10 = Σ b_r x^r where b_r is the coefficient of x^r in this expansion.</p><p>From Step 2: (2 + 3x + 4x²)^10 expanded gives us Σ a_r x^(20-r)</p><p>Therefore, the coefficient of x^k in (2 + 3x + 4x²)^10 equals a_(20-k).</p><p>This means: coefficient of x^13 in (2 + 3x + 4x²)^10 = a_7</p><p>And: coefficient of x^7 in (2 + 3x + 4x²)^10 = a_13</p><p><strong>Step 4: Establish the ratio</strong></p><p>Let c_r = coefficient of x^r in (2 + 3x + 4x²)^10</p><p>Then: a_7 = c_13 and a_13 = c_7</p><p><strong>Step 5: Use another substitution</strong></p><p>Multiply the original by x^(-20): x^(-20)(2x² + 3x + 4)^10 = Σ a_r x^(r-20)</p><p>Setting x = 1: (2 + 3 + 4)^10 = Σ a_r, so 9^10 = Σ a_r (sum of all coefficients)</p><p><strong>Step 6: Apply symmetry directly</strong></p><p>Notice: 2x² + 3x + 4 and its "reciprocal" form 4x² + 3x + 2 have complementary coefficients.</p><p>The coefficient of x^r in (2x² + 3x + 4)^10 divided by the coefficient of x^(20-r) in (4x² + 3x + 2)^10 relates through the leading/trailing coefficients ratio.</p><p>For (2x² + 3x + 4)^10 vs (4x² + 3x + 2)^10:</p><p>a_r/a_(20-r) = (2/4)^10 = (1/2)^10 = 1/1024</p><p><strong>Step 7: Calculate the ratio</strong></p><p>a_7/a_13 = a_7/a_(20-7) = (1/2)^10 = 1/1024</p><p><strong>∴ Answer: 1/1024</strong></p>
Correct Answer: 1