Answer :
Final answer:
The binomial expansion of (2y + 4x³)⁴ can be found using the binomial theorem. The coefficient of the x⁹ term is found by substituting a=2y, b=4x³, and n=4 into the theorem and solving for k=3, resulting in a coefficient of 32.
Explanation:
The binomial expansion of a quantity represents its expansion into an infinite series of terms. In this case, the binomial is (2y + 4x³)⁴. To find the coefficient of the x⁹ term, we use the binomial theorem, which states that (a + b)ⁿ = Σ[k=0, n] (n choose k) * aⁿ-k * bᵏ. In this equation, 'n choose k' represents the number of ways you can choose k items from a set of n.
Applying this to your equation, a = 2y, b = 4x³, and n = 4. We want to find k when bᵏ = x⁹. Because b = 4x³, we have (4x³)ᵏ = x⁹, which simplifies to 4ᵏ * x⁽³ᵏ⁾ = x⁹. This indicates k = 3. Substituting these values into the binomial theorem, the coefficient of the x⁹ term is (4 choose 3)*(2y)⁴⁻³*(4)³, which simplifies to 4*2x³ = 32.
Learn more about Binomial Expansion here:
https://brainly.com/question/12249986
#SPJ11
