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Chapter 10: Problem 55

Determine whether each statement is true or false. The coefficient of \(x^{8}\) in the expansion of \((2 x-1)^{12}\) is 126,720

Short Answer

Expert verified
True, the coefficient of \(x^8\) is 126,720.

Step by step solution

01

Identify the general term formula

In the binomial expansion of \((a + b)^n\), the general term is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). For our given expression \((2x - 1)^{12}\), \(a = 2x\), \(b = -1\), and \(n = 12\).
02

Find the specific term for \(x^8\)

We need the term where the power of \(x\) is 8. In the expression \((2x)^{12-k}(-1)^k\), \((2x)^{12-k}\) should contribute to \(x^8\). This means we need \(12 - k = 8\), giving \(k = 4\).
03

Calculate the coefficient for the term

Substitute \(k = 4\) into the general term formula: \[T_{5} = \binom{12}{4} (2x)^{12-4} (-1)^{4}\]
04

Compute the binomial coefficient

Calculate \(\binom{12}{4}\): \[\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495\]
05

Compute the power of each factor

Compute \((2x)^8\) which contributes to the term: \((2)^8 = 256\) and \((-1)^4 = 1\).
06

Determine the full coefficient

Multiply the calculated binomial coefficient by the powers: \[495 \times 256 = 126,720\]. Thus, the coefficient of \(x^8\) is indeed 126,720.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Theorem
The binomial theorem is a fundamental principle in algebra that allows us to expand expressions raised to a power. It provides a formula to expand expressions of the form \( (a + b)^n \). The theorem states that each term in the expansion is a result of a combination of coefficients, powers of \(a\), and powers of \(b\). To elaborate:
  • It breaks down the expression \((a+b)^n\) into a series of terms.
  • Each term is formed by selecting \(k\) elements to multiply \(a\) and \(k\) to multiply \(b\).
  • The combination of these selections is denoted by the binomial coefficient \(\binom{n}{k}\).
This allows for the seamless expansion of polynomials, making difficult computations easier to manage and understand.
Binomial Coefficient
The binomial coefficient is a key component in the binomial expansion and is symbolized as \(\binom{n}{k}\). This term represents the number of ways to choose \(k\) items from \(n\) items, without considering the order, often described as "n choose k." The mathematical formula for calculating a binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(!\) denotes factorial, meaning the product of all positive integers up to that number. For instance,
  • \(n!\) means \(n \times (n-1) \times (n-2) \times \ldots \times 1\).
  • Using this formula, the binomial coefficient helps determine the weight each term holds in the expansion.
In the problem, \(\binom{12}{4}\) is calculated to find the coefficient for the term required in the expansion.
Polynomial Expansion
Polynomial expansion involves expressing a power of a binomial, such as \((a + b)^n\), into a sum of multiple terms. Each term in this sum is a product of powers of \(a\) and \(b\), with coefficients determined by the binomial coefficient. Let's break this down:
  • Each term has a specific power of \(x\), determined by the binomial theorem.
  • In the given exercise, the expansion of \((2x - 1)^{12}\) was computed to find the specific term for \(x^8\).
  • This was achieved by setting \(12-k = 8\), which gave us \(k = 4\). This means the required term is influenced by \((2x)^8\).
      • These expansions are practical for solving equations, finding terms, or coefficients, and even in calculus for approximation methods. Polynomial expansions transform products into manageable sums, simplifying complex computations.

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