Use Pascal's Triangle to expand the binomial $(3 x-4)^3$.
A. $27 x^3+108 x^2+144 x+64$
B. $27 x^3+108 x^2+144 x-64$
C. $27 x^3-108 x^2+144 x-64$
D. $27 x^3-108 x^2-144 x+64$
Answer (2)
Recognize the binomial expansion pattern using Pascal's Triangle.
Substitute a = 3 x and b = − 4 into the binomial expansion formula: ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 .
Calculate each term of the expansion.
Combine the terms to obtain the final expanded form: 27 x 3 − 108 x 2 + 144 x − 64 .
Explanation
Understanding the Problem We are asked to expand the binomial ( 3 x − 4 ) 3 using Pascal's Triangle. Pascal's Triangle gives us the coefficients for the expansion of a binomial raised to a power.
Binomial Expansion The binomial expansion of ( a + b ) 3 is given by a 3 + 3 a 2 b + 3 a b 2 + b 3 . This corresponds to the coefficients 1, 3, 3, 1 from Pascal's Triangle.
Substitution In our case, a = 3 x and b = − 4 . Substituting these values into the expansion, we get: ( 3 x ) 3 + 3 ( 3 x ) 2 ( − 4 ) + 3 ( 3 x ) ( − 4 ) 2 + ( − 4 ) 3
Calculation Now, let's calculate each term:
( 3 x ) 3 = 27 x 3
3 ( 3 x ) 2 ( − 4 ) = 3 ( 9 x 2 ) ( − 4 ) = − 108 x 2
3 ( 3 x ) ( − 4 ) 2 = 3 ( 3 x ) ( 16 ) = 144 x
( − 4 ) 3 = − 64
So the expansion is 27 x 3 − 108 x 2 + 144 x − 64 .
Final Answer Therefore, the expansion of ( 3 x − 4 ) 3 is 27 x 3 − 108 x 2 + 144 x − 64 .
Examples
Binomial expansions are used in various fields such as probability, statistics, and calculus. For example, in probability, when calculating the probability of a certain number of successes in a series of independent trials, binomial expansion can be used. In engineering, binomial expansions can approximate complex functions, simplifying calculations and providing insights into system behavior. They also appear in physics, such as in quantum mechanics when dealing with perturbation theory.
Using the binomial expansion formula, the expansion of ( 3 x − 4 ) 3 leads to the result 27 x 3 − 108 x 2 + 144 x − 64 . Thus, the answer is option C. This is calculated by substituting values into the binomial expansion and simplifying the terms.
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