Problem 204 Expand each binomial using Pasca... [FREE SOLUTION]

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Chapter 12: Problem 204

Expand each binomial using Pascal's Triangle. \((4 x-1)^{3}\)

Short Answer

Expert verified

The expanded form is \( 64x^3 - 48x^2 + 12x - 1 \).

Step by step solution

Identify the binomial

The given binomial is \( (4x - 1)^3 \).

Write down Pascal's Triangle row

For an exponent of 3, the Pascal's Triangle row is: 1, 3, 3, 1.

Apply the binomial theorem

The binomial theorem states \[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k}b^k \]. Apply this for \( a = 4x \) and \( b = -1 \).

Expand using coefficients from Pascal's Triangle

Using the coefficients 1, 3, 3, 1, expand as follows: \( 1(4x)^3(-1)^0 + 3(4x)^2(-1)^1 + 3(4x)^1(-1)^2 + 1(4x)^0(-1)^3 \).

Simplify each term

Calculate each term in the expansion: \ \(1 \times 64x^3 \times 1 + 3 \times 16x^2 \times (-1) + 3 \times 4x \times 1 + 1 \times 1 \times (-1) \).

Combine the terms

Combine all simplified terms: \( 64x^3 - 48x^2 + 12x - 1 \).

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Key Concepts

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

binomial expansion

Binomial expansion is the process of expanding the power of a binomial expression. A binomial is an algebraic expression containing two terms, such as \(a + b\). When we raise a binomial to a power, we can expand it to express it as a sum of several terms. For example, \( (a + b)^3 \) expands to \(a^3 + 3a^2b + 3ab^2 + b^3\). The coefficients of these terms are derived using the binomial theorem, which can be facilitated by Pascal's Triangle.
Binomial expansion is useful in Algebra for simplifying expressions and solving polynomial equations. It provides a structured way to handle powers of binomials without multiplying the terms repeatedly.

binomial theorem

The binomial theorem states that any binomial expression raised to a power \(n\) can be expanded into a sum involving terms of the form \(a^k \cdot b^{n-k}\), where the coefficients follow the binomial coefficients 鈥� commonly represented by \( \binom{n}{k} \). Formally, the binomial theorem is expressed as:
\[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k}b^k \]
Here, \( \binom{n}{k} \) represents the binomial coefficient and is calculated as \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \].
This theorem helps us expand expressions like \( (4x - 1)^3 \) by providing a systematic method to determine each term in the expansion. Key points to note:

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91影视

Expand each binomial using Pascal's Triangle. \((4 x-1)^{3}\)

Short Answer

Step by step solution

Identify the binomial

Write down Pascal's Triangle row

Apply the binomial theorem

Expand using coefficients from Pascal's Triangle

Simplify each term

Combine the terms

Key Concepts

binomial expansion

binomial theorem

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