Find the middle term in the expansion of $(2 x-y)^4$ and simplify your answer.
Find the term that contains $x^5$ in the expansion of $(2 x+3 y)^8$.
Answer (1)
Find the middle term in ( 2 x − y ) 4 using the binomial theorem with k = 2 : ( 2 4 ) ( 2 x ) 2 ( − y ) 2 = 6 ( 4 x 2 ) ( y 2 ) .
Simplify the middle term: 24 x 2 y 2 .
Find the term containing x 5 in ( 2 x + 3 y ) 8 using the binomial theorem with k = 3 : ( 3 8 ) ( 2 x ) 5 ( 3 y ) 3 = 56 ( 32 x 5 ) ( 27 y 3 ) .
Simplify the term containing x 5 : 48384 x 5 y 3 .
24 x 2 y 2 and 48384 x 5 y 3
Explanation
Understanding the Problem We are asked to find the middle term in the expansion of ( 2 x − y ) 4 and the term containing x 5 in the expansion of ( 2 x + 3 y ) 8 . We will use the binomial theorem to solve both parts of the problem. The binomial theorem states that ( a + b ) n = ∑ k = 0 n ( k n ) a n − k b k .
Finding the Middle Term For part 4.1, we need to find the middle term in the expansion of ( 2 x − y ) 4 . Since the power is 4, there are 5 terms in the expansion (corresponding to k = 0 , 1 , 2 , 3 , 4 ). The middle term is the third term, which corresponds to k = 2 in the binomial expansion.
Applying the Binomial Theorem We apply the binomial theorem to find the middle term: ( 2 4 ) ( 2 x ) 4 − 2 ( − y ) 2
Simplifying the Expression We simplify the expression. We know that ( 2 4 ) = 6 . So, we have: 6 ( 2 x ) 2 ( − y ) 2 = 6 ( 4 x 2 ) ( y 2 ) = 24 x 2 y 2
Finding the Term with x^5 For part 4.2, we need to find the term that contains x 5 in the expansion of ( 2 x + 3 y ) 8 . We use the binomial theorem to express the general term in the expansion of ( 2 x + 3 y ) 8 as ( k 8 ) ( 2 x ) 8 − k ( 3 y ) k
Determining the Value of k We need to find the value of k such that the term contains x 5 . This means 8 − k = 5 , so k = 3 .
Substituting k into the General Term We substitute k = 3 into the general term: ( 3 8 ) ( 2 x ) 8 − 3 ( 3 y ) 3 = ( 3 8 ) ( 2 x ) 5 ( 3 y ) 3
Simplifying the Expression We simplify the expression. We know that ( 3 8 ) = 56 . So, we have: 56 ( 2 x ) 5 ( 3 y ) 3 = 56 ( 32 x 5 ) ( 27 y 3 ) = 56 × 32 × 27 x 5 y 3 = 48384 x 5 y 3
Final Answer Therefore, the middle term in the expansion of ( 2 x − y ) 4 is 24 x 2 y 2 , and the term containing x 5 in the expansion of ( 2 x + 3 y ) 8 is 48384 x 5 y 3 .
Examples
Binomial expansions are used in various fields such as probability, statistics, and physics. For example, in probability, the binomial theorem can be used to calculate the probability of a certain number of successes in a series of independent trials. In physics, it can be used to approximate complex expressions. Understanding binomial expansions helps in modeling and solving real-world problems involving combinations and probabilities.
