Using the binomial theorem, what is the expansion of $(x+3y)^5$?
A. $(x+3 y)^5=x^5+15 x^4 y+90 x^3 y^2+270 x^2 y^3+405 x y^4+243 y^5$
B. $(x+3 y)^5=x^5+5 x^4 y+10 x^3 y^2+10 x^2 y^3+5 y^4+y^5$
C. $(x+3 y)^5=x^5+x^4 y+x^3 y^2+x^2 y^3+x y^4+y^5$
D. $(x+3 y)^5=y^5+15 y^4 x+90 y^3 x^2+270 y^2 x^3+405 y x^4+243 x^5
Answer (2)
Apply the binomial theorem to expand ( x + 3 y ) 5 .
Calculate the binomial coefficients and powers of 3 y .
Substitute the calculated values into the expansion.
Simplify the expression to obtain the correct expansion: ( x + 3 y ) 5 = x 5 + 15 x 4 y + 90 x 3 y 2 + 270 x 2 y 3 + 405 x y 4 + 243 y 5 .
Explanation
Understanding the Problem We are asked to identify the correct binomial expansion of ( x + 3 y ) 5 from the given options. The binomial theorem provides a formula for expanding expressions of the form ( a + b ) n .
Applying the Binomial Theorem The binomial theorem states that ( a + b ) n = ∑ k = 0 n ( k n ) a n − k b k , where ( k n ) = k ! ( n − k )! n ! . In our case, a = x , b = 3 y , and n = 5 . So we need to expand ( x + 3 y ) 5 using this theorem.
Expanding the Expression The expansion will be of the form: ( x + 3 y ) 5 = ( 0 5 ) x 5 ( 3 y ) 0 + ( 1 5 ) x 4 ( 3 y ) 1 + ( 2 5 ) x 3 ( 3 y ) 2 + ( 3 5 ) x 2 ( 3 y ) 3 + ( 4 5 ) x 1 ( 3 y ) 4 + ( 5 5 ) x 0 ( 3 y ) 5
Calculating Binomial Coefficients Let's calculate the binomial coefficients: ( 0 5 ) = 1 ( 1 5 ) = 5 ( 2 5 ) = 2 ! 3 ! 5 ! = 2 5 × 4 = 10 ( 3 5 ) = 3 ! 2 ! 5 ! = 2 5 × 4 = 10 ( 4 5 ) = 5 ( 5 5 ) = 1
Calculating Powers of 3y Now, let's calculate the powers of 3 y :
( 3 y ) 0 = 1 ( 3 y ) 1 = 3 y ( 3 y ) 2 = 9 y 2 ( 3 y ) 3 = 27 y 3 ( 3 y ) 4 = 81 y 4 ( 3 y ) 5 = 243 y 5
Substituting Values Substitute these values back into the expansion: ( x + 3 y ) 5 = 1 ⋅ x 5 ⋅ 1 + 5 ⋅ x 4 ⋅ 3 y + 10 ⋅ x 3 ⋅ 9 y 2 + 10 ⋅ x 2 ⋅ 27 y 3 + 5 ⋅ x ⋅ 81 y 4 + 1 ⋅ 1 ⋅ 243 y 5
Simplifying the Expression Simplify the expression: ( x + 3 y ) 5 = x 5 + 15 x 4 y + 90 x 3 y 2 + 270 x 2 y 3 + 405 x y 4 + 243 y 5
Identifying the Correct Expansion Comparing the derived expansion with the given options, we find that the correct expansion is: ( x + 3 y ) 5 = x 5 + 15 x 4 y + 90 x 3 y 2 + 270 x 2 y 3 + 405 x y 4 + 243 y 5
Examples
The binomial theorem is not just an abstract mathematical concept; it has practical applications in various fields. For instance, in finance, it can be used to model investment growth with varying interest rates. Imagine you invest an amount that grows at a certain rate, but this rate fluctuates. The binomial theorem helps in calculating the probabilities of different growth scenarios, allowing for more informed investment decisions. It also finds use in physics for approximating complex equations and in computer science for algorithm analysis.
Using the binomial theorem, the expansion of ( x + 3 y ) 5 is found to be x 5 + 15 x 4 y + 90 x 3 y 2 + 270 x 2 y 3 + 405 x y 4 + 243 y 5 , matching option A.
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