Find the middle term in the expansion of $(2 x-y)^4$ and simplify your answer.
Answer (1)
Identify the middle term in the binomial expansion, which is the third term when n = 4 .
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 and simplify.
The middle term is 24 x 2 y 2 .
Explanation
Identifying 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 will be the third term.
Applying 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 . Since we want the third term, we set k = 2 .
Calculating the Binomial Coefficient and Powers Substituting n = 4 and k = 2 into the general term formula, we get:
T 2 + 1 = T 3 = ( 2 4 ) ( 2 x ) 4 − 2 ( − y ) 2
We know that ( 2 4 ) = 2 ! 2 ! 4 ! = ( 2 × 1 ) ( 2 × 1 ) 4 × 3 × 2 × 1 = 6 . Also, ( 2 x ) 4 − 2 = ( 2 x ) 2 = 4 x 2 , and ( − y ) 2 = y 2 .
Simplifying the Expression Therefore, the middle term is:
T 3 = 6 ( 4 x 2 ) ( y 2 ) = 24 x 2 y 2
Final Answer The middle term in the expansion of ( 2 x − y ) 4 is 24 x 2 y 2 .
Examples
Binomial expansions are used in probability calculations, such as determining the likelihood of specific outcomes in a series of independent trials. For example, if you flip a coin four times, the expansion of ( p + q ) 4 , where p is the probability of heads and q is the probability of tails, can help you calculate the probability of getting exactly two heads.
