Use the binomial theorem expression to find the expansion of $(2 x+1)^4$.
A. $(2 x+1)^4=32 x^4+8 x^3+16 x^2+x+8$
B. $(2 x+1)^4=x^4+4 x^3+6 x^2+4 x+1$
C. $(2 x+1)^4=16 x^4+32 x^3+24 x^2+8 x+1$
D. $(2 x+1)^4=x^4+x^3+x^2+x
Answer (2)
Apply the binomial theorem to expand ( 2 x + 1 ) 4 .
Calculate the binomial coefficients.
Substitute the binomial coefficients into the expansion and simplify the terms.
The final expansion is 16 x 4 + 32 x 3 + 24 x 2 + 8 x + 1 .
Explanation
Understanding the Problem We want to find the expansion of ( 2 x + 1 ) 4 using the binomial theorem.
Stating the Binomial Theorem The binomial theorem states that ( a + b ) n = ∑ k = 0 n ( k n ) a n − k b k
Applying the Theorem Applying the binomial theorem to expand ( 2 x + 1 ) 4 , we have: ( 2 x + 1 ) 4 = ( 0 4 ) ( 2 x ) 4 ( 1 ) 0 + ( 1 4 ) ( 2 x ) 3 ( 1 ) 1 + ( 2 4 ) ( 2 x ) 2 ( 1 ) 2 + ( 3 4 ) ( 2 x ) 1 ( 1 ) 3 + ( 4 4 ) ( 2 x ) 0 ( 1 ) 4
Calculating Binomial Coefficients Now, let's calculate the binomial coefficients: ( 0 4 ) = 1 , ( 1 4 ) = 4 , ( 2 4 ) = 6 , ( 3 4 ) = 4 , ( 4 4 ) = 1
Substituting Coefficients Substitute the binomial coefficients into the expansion: ( 2 x + 1 ) 4 = 1 ( 2 x ) 4 ( 1 ) 0 + 4 ( 2 x ) 3 ( 1 ) 1 + 6 ( 2 x ) 2 ( 1 ) 2 + 4 ( 2 x ) 1 ( 1 ) 3 + 1 ( 2 x ) 0 ( 1 ) 4
Simplifying Terms Simplify the terms: ( 2 x + 1 ) 4 = 1 ( 16 x 4 ) ( 1 ) + 4 ( 8 x 3 ) ( 1 ) + 6 ( 4 x 2 ) ( 1 ) + 4 ( 2 x ) ( 1 ) + 1 ( 1 ) ( 1 )
Calculating the Expansion Finally, calculate the final expansion: ( 2 x + 1 ) 4 = 16 x 4 + 32 x 3 + 24 x 2 + 8 x + 1
Final Answer Therefore, the expansion of ( 2 x + 1 ) 4 is 16 x 4 + 32 x 3 + 24 x 2 + 8 x + 1
Examples
The binomial theorem is not just an abstract mathematical concept; it has practical applications in various fields. For instance, it can be used to model the growth of populations or investments over time. Imagine you're calculating the future value of an investment that grows at a certain percentage annually. The binomial theorem helps break down the compounding effect, allowing you to predict the outcome more accurately. It's also used in probability theory to calculate the likelihood of different outcomes in repeated trials, such as coin flips or manufacturing quality control. Understanding the binomial theorem provides a powerful tool for analyzing and predicting outcomes in diverse real-world scenarios.
The expansion of ( 2 x + 1 ) 4 using the binomial theorem results in 16 x 4 + 32 x 3 + 24 x 2 + 8 x + 1 . This corresponds to option C in the provided choices. The solution involves applying the binomial theorem to calculate each term's contribution based on the binomial coefficients.
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