Find the middle term in the expansion of $(2 x-y)^4$ and simplify your answer.
Answer (1)
Identify the middle term in the expansion of ( 2 x − y ) 4 as the third term.
Apply the binomial theorem to find the general term: T k + 1 = ( k n ) a n − k b k .
Substitute n = 4 , k = 2 , a = 2 x , and b = − y into the formula.
Simplify the expression to get the middle term: 24 x 2 y 2 .
Explanation
Finding the Middle Term We are asked to find the middle term in the expansion of ( 2 x − y ) 4 . The binomial expansion will have 4 + 1 = 5 terms. Therefore, the middle term is the third term.
Using the Binomial Theorem The general term in the binomial expansion of ( a + b ) n is given by T k + 1 = ( k n ) a n − k b k , where k = 0 , 1 , 2 , ... , n . In our case, a = 2 x , b = − y , and n = 4 . We want to find the third term, so we set k = 2 .
Substituting the Values Now, we substitute the values into the formula: T 2 + 1 = T 3 = ( 2 4 ) ( 2 x ) 4 − 2 ( − y ) 2 .
Calculating the Binomial Coefficient We calculate the binomial coefficient: ( 2 4 ) = 2 ! ( 4 − 2 )! 4 ! = 2 ! 2 ! 4 ! = ( 2 × 1 ) ( 2 × 1 ) 4 × 3 × 2 × 1 = 4 24 = 6 .
Simplifying the Expression Next, we simplify the expression: ( 2 x ) 4 − 2 = ( 2 x ) 2 = 4 x 2 and ( − y ) 2 = y 2 .
Finding the Middle Term Now, we multiply all the terms together: T 3 = 6 × 4 x 2 × y 2 = 24 x 2 y 2 .
Examples
Binomial expansions are used in probability calculations, such as determining the likelihood of different outcomes in a series of independent trials. For instance, if you flip a coin four times, the expansion of ( H + T ) 4 can help you calculate the probability of getting a specific number of heads and tails. This concept extends to various fields like genetics, finance, and engineering, where understanding the probabilities of different events is crucial for decision-making.
