QUESTION 4
4.1 Find the middle term in the expansion of $(2 x-y)^4$ and simplify your answer.
(4)
4.2 Find the term that contains $x^5$ in the expansion of $(2 x+3 y)^8$.
(4)
[8]
TOTAL: [50]
Answer (2)
The middle term in the expansion of ( 2 x − y ) 4 is 24 x 2 y 2 , and the term that contains x 5 in the expansion of ( 2 x + 3 y ) 8 is 48384 x 5 y 3 .
;
Find the middle term in the expansion of ( 2 x − y ) 4 using the binomial theorem: ( 2 4 ) ( 2 x ) 2 ( − y ) 2 = 24 x 2 y 2 .
Find the term containing x 5 in the expansion of ( 2 x + 3 y ) 8 using the binomial theorem: ( 3 8 ) ( 2 x ) 5 ( 3 y ) 3 = 48384 x 5 y 3 .
The middle term is 24 x 2 y 2 .
The term containing x 5 is 48384 x 5 y 3 , so the final answer is 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 that contains x 5 in the expansion of ( 2 x + 3 y ) 8 . We will use the binomial theorem to solve this 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. The middle term is the third term, which corresponds to k = 2 in the binomial expansion. Using the binomial theorem, the middle term is given by: ( 2 4 ) ( 2 x ) 4 − 2 ( − y ) 2 = ( 2 4 ) ( 2 x ) 2 ( − y ) 2 We know that ( 2 4 ) = 2 ! 2 ! 4 ! = 2 × 1 4 × 3 = 6 . Therefore, the middle term is: 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 . Using the binomial theorem, the general term in the expansion of ( 2 x + 3 y ) 8 is given by: ( k 8 ) ( 2 x ) 8 − k ( 3 y ) k We want to find the value of k such that the term contains x 5 . This means 8 − k = 5 , so k = 3 . Substituting k = 3 into the general term, we get: ( 3 8 ) ( 2 x ) 8 − 3 ( 3 y ) 3 = ( 3 8 ) ( 2 x ) 5 ( 3 y ) 3 We know that ( 3 8 ) = 3 ! 5 ! 8 ! = 3 × 2 × 1 8 × 7 × 6 = 56 . Therefore, the term containing x 5 is: 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 that contains 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, they can be used to calculate the probability of a certain number of successes in a series of independent trials. In physics, they can be used to approximate complex equations and simplify calculations. Understanding binomial expansions allows us to model and analyze real-world phenomena more effectively.
