Find the term that contains $x^5$ in the expansion of $(2 x+3 y)^8$.
Answer (1)
Use the binomial theorem to expand ( 2 x + 3 y ) 8 .
Identify the term containing x 5 by setting 8 − k = 5 , which gives k = 3 .
Calculate the binomial coefficient ( 3 8 ) = 56 .
Compute the term: 56 × ( 2 x ) 5 × ( 3 y ) 3 = 48384 x 5 y 3 . The term containing x 5 is 48384 x 5 y 3 .
Explanation
Problem Analysis We are given the binomial expression ( 2 x + 3 y ) 8 and we want to find the term that contains x 5 in its expansion. We will use the binomial theorem to solve this problem.
Binomial Theorem The binomial theorem states that for any non-negative integer n , the expansion of ( a + b ) n is given by: ( a + b ) n = k = 0 ∑ n ( k n ) a n − k b k In our case, a = 2 x , b = 3 y , and n = 8 .
Finding the value of k 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 term with x 5 , so we need to find the term where 8 − k = 5 . This implies that k = 8 − 5 = 3 .
Calculating the Term Now, we substitute k = 3 into the general term to find the specific term containing x 5 :
( 3 8 ) ( 2 x ) 8 − 3 ( 3 y ) 3 = ( 3 8 ) ( 2 x ) 5 ( 3 y ) 3 We calculate the binomial coefficient ( 3 8 ) :
( 3 8 ) = 3 ! ( 8 − 3 )! 8 ! = 3 ! 5 ! 8 ! = 3 × 2 × 1 8 × 7 × 6 = 56 We also calculate ( 2 x ) 5 and ( 3 y ) 3 :
( 2 x ) 5 = 2 5 x 5 = 32 x 5 ( 3 y ) 3 = 3 3 y 3 = 27 y 3
Final Calculation Now, we multiply the coefficients together: 56 × 32 × 27 = 48384 Therefore, the term containing x 5 in the expansion of ( 2 x + 3 y ) 8 is 48384 x 5 y 3 .
Final Answer The term containing x 5 in the expansion of ( 2 x + 3 y ) 8 is 48384 x 5 y 3 .
Examples
The binomial theorem is not just an abstract mathematical concept; it has practical applications in various fields. For instance, in probability, it helps calculate the likelihood of specific outcomes in a series of independent trials, like coin flips or dice rolls. In finance, it can be used to model investment growth and predict future values based on certain probabilities. Moreover, in physics, the binomial theorem appears in approximations and expansions used to simplify complex equations, making it a versatile tool for problem-solving across disciplines. Understanding the binomial theorem provides a foundation for tackling real-world scenarios involving probabilities, predictions, and approximations.
