Find the coefficient "a" of the given term in the expansion of the binomial.

Question

Binomial: (4x - y)¹²

Term: ax³y⁹

The problem also needs to be a a large number but i have no idea how to get to that point

Answer 1

We can typically expect a "12th power" polynomial to have 13 terms. Each term includes a contribution from the first term in the binomial, in this case "4x" (let's call the coefficient "a"), the last term in the binomial, in this case "-1y" (let's call the coefficient "b"), and from Pascal's triangle (let's call that "p"). A 12th degree polynomial (with first binomial variable "x" and last variable "y") would generically look like this:

p₁a₁₂x¹²b₀y⁰ + p₂a₁₁x¹¹b₁y¹ + p₃a₁₀x¹⁰b₂y² + p₄a₉x⁹b₃y³ + p₅a₈x⁸b₄y⁴ + p₆a₇x⁷b₅y⁵ + p₇a₆x⁶b₆y⁶ + p₈a₅x⁵b₇y⁷ + p₉a₄x⁴b₈y⁸ + p₁₀a₃x³b₉y⁹ + p₁₁a₂x²b₁₀y¹⁰ + p₁₂a₁x¹b₁₁y¹¹ + p₁₃a₀x⁰b₁₂y¹²

So the term we are looking for is what I have listed as p₁₀a₃x³b₉y⁹ so we just need p₁₀, a₃, and b₉. The coefficient we are seeking will just be the product of those three.

In this case, the coefficient of the first term in the binomial is "4" so a₃ will be 4³ because the "x" variable is raised to the 3rd power. So a₃ = 64

In this case, the coefficient of the last term in the binomial is "-1" so b₉ will be (-1)⁹ because the "y" variable is raised to the 9^th power. So b₉ = -1.

Using the 12th row on Pascal's triangle, we get:

1         12       66       220     495     792     924     792     495     220     66       12       1

We can see that p₁₀ = 220.

So the coefficient we seek is (64)(-1)(220) = -14080