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

Question

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

Binomial: (4x − y)¹²

Term: ax³y⁹

a=

Answer 1

To find the coefficient of any term in this expansion, you can use the binomial theorem. The binomial theorem formula says that

(a+b)ⁿ = _k=0ⁿ∑(ⁿ_k)a^kb^n-k

In this case we have (4x-y)¹² so if we compare this to the above formula for the binomial expansion, we can say that:

a = 4x

b = -y

n = 12

The term we care about is the x³y⁹ term. This term will occur when k = 3. So if we look at the term in the binomial theorem formula that occurs when k=3, that would tell us what that entire term would be.

(ⁿ_k)a^kb^n-k

(¹²₃)(4x)³(-y)⁹

Then you just need to simplify from there to figure out the coefficient of this term. Note that (¹²₃) "12 choose 3" can be computed with a calculator or by using the formula for combinations: (_ⁿr) = (n!)/(r!(n-r)!)