Problem 72 - Algebra 2

Question

Expand each binomial. $$ (x-2)^{4} $$

Step-by-Step Solution

Answer

The expanded form of \((x-2)^4\) is \(x^4 - 8x^3 + 24x^2 - 32x + 16\).

Step 1: Identify the Binomial Theorem formula

Binomial Theorem formula is given by \[(a+b)^n = \sum_{k=0}^n {n \choose k}a^{n-k}b^k\] where \(n \choose k \) represents the binomial coefficients, and 'a' and 'b' are the elements of the binomial.

Step 2: Substitute values into the formula

In the binomial \((x-2)^4\), 'a' is 'x', 'b' is '-2', and 'n' is 4. Substituting these values into the Binomial theorem yields \[(x-2)^4 = \sum_{k=0}^4 {4 \choose k}x^{4-k}(-2)^k\].

Step 3: Calculate individual terms

To obtain all the terms in the expansion, determine each term when 'k' equals 0 to 4. The result is \[(x-2)^4 = {4 \choose 0}x^4*(-2)^0 + {4 \choose 1}x^3*(-2)^1 + {4 \choose 2}x^2*(-2)^2 + {4 \choose 3}x^1*(-2)^3 + {4 \choose 4}x^0*(-2)^4 = x^4 - 8x^3 + 24x^2 - 32x + 16\]

Step 4: Compose Final Polynomial

The expanded form of the binomial is the sum of all the calculated terms, i.e., \(x^4 - 8x^3 + 24x^2 - 32x + 16\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a fundamental concept in algebra that involves expressing a power of a binomial expression in its expanded polynomial form. Specifically, expanding a binomial refers to representing it as the sum of individual terms rather than keeping it in the compact power form.
For instance, consider the binomial \( (x-2)^4 \). Instead of multiplying \((x-2)\) four times, we use the Binomial Theorem to transform it into a series of terms joined by addition and/or subtraction. This expansion involves exponents descending from the power of the binomial on the first element, while the second element's exponents ascend.
Understanding polynomial expansions allows for simplifying complex algebraic expressions, which is crucial for solving equations in calculus, physics, and engineering. It breaks down expressions into manageable parts, often simplifying them for further analysis or computation.

Binomial Coefficients

Binomial coefficients are numbers that play a pivotal role in the expansion of binomials. They are denoted as \({n \choose k}\), read as 'n choose k', and are found in Pascal’s Triangle. These coefficients determine the multiplier of each term in a binomial expansion.
For the expression \((x-2)^4\), the coefficients are given by

• \({4 \choose 0} = 1\)
• \({4 \choose 1} = 4\)
• \({4 \choose 2} = 6\)
• \({4 \choose 3} = 4\)
• \({4 \choose 4} = 1\)

Each term in the expanded polynomial is the product of a binomial coefficient and the variables with specific powers. These coefficients ensure that each term in the polynomial equally contributes to the accuracy of the expansion, maintaining the symmetry between the number of terms and the degree of the polynomial.

Binomial Expression

A binomial expression is a polynomial with exactly two terms. Commonly represented as \((a+b)\), it can consist of either numbers or variables with an operation between them, such as addition or subtraction.
In the example \((x-2)^4\), the binomial expression is \((x-2)\). Here, \(x\) and \(-2\) are the two parts of the binomial. Binomial expressions are foundational in algebra because they often appear in various mathematical problems and scenarios.
When raised to a power, binomial expressions are expanded using the Binomial Theorem, allowing us to analyze and simplify higher degree polynomials. These expansions enable deeper insights into roots, solutions, and functions related to the algebraic expressions.