Problem 107 - Intermediate Algebra

Question

Perform each indicated operation. See Sections 1.4 and 5.4 $$ (x-4)^{2} $$

Step-by-Step Solution

Answer

The expression \((x-4)^{2}\) simplifies to \(x^{2} - 8x + 16\).

Step 1: Understand the Expression

The given expression is \((x-4)^{2}\). This is a binomial raised to the power of 2.

Step 2: Apply the Binomial Square Formula

Use the formula \((a-b)^{2} = a^{2} - 2ab + b^{2}\). Here, \(a = x\) and \(b = 4\).

Step 3: Calculate \(a^2\)

Using the formula, calculate \(a^{2} = x^{2}\).

Step 4: Calculate \(-2ab\)

Calculate \(-2ab = -2(x)(4) = -8x\).

Step 5: Calculate \(b^2\)

Calculate \(b^{2} = 4^{2} = 16\).

Step 6: Combine All Terms

Combine the terms calculated: \(x^{2} - 8x + 16\).

Key Concepts

Understanding Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power. Typically, it involves expanding expressions of the form \((a + b)^n\), where \(n\) is a non-negative integer. This expansion involves using binomial coefficients that multiply the terms in a specific pattern known as the binomial theorem. For small powers, such as 2, we can use specific formulas like the binomial square formula to make expansion simple. The binomial expansion helps simplify expressions and is a foundational concept in algebra. It is especially important in higher mathematics and is commonly used to solve polynomial equations.Some key points to remember about binomial expansion include:

• It must involve a binomial term (two terms) raised to a power.
• Each term in the expansion is a product of a coefficient, a power of the first term, and a power of the second term.
• The sum of the powers of the two terms in each expanded term always equals \(n\).

The Binomial Square Formula

The binomial square formula is a special case of the binomial expansion. It simplifies the process of squaring a binomial expression into a quadratic polynomial. This formula is specifically for expressions where the power is 2, such as \((a - b)^2\). According to the formula: \[(a-b)^2 = a^2 - 2ab + b^2\] This formula makes expanding expressions like \((x-4)^2\) much easier. Here's how it works:

• Calculate \(a^2\): For our expression, \(a = x\) so \(a^2 = x^2\).
• Calculate \(-2ab\): This results in \(-2(x)(4) = -8x\).
• Calculate \(b^2\): Here, \(b = 4\) so \(b^2 = 16\).

By combining these results, you obtain the expanded form: \(x^2 - 8x + 16\). This formula is easy to memorize and apply for quick expansion of squared binomials.

Algebraic Expressions and Their Importance

Algebraic expressions are combinations of constants, variables, and operations (such as addition, subtraction, multiplication) that represent values. They form the basis of algebra and are used to model real-world situations, solve problems, and explore mathematical relationships. In understanding how to work with algebraic expressions, it is vital to comprehend how operations affect terms and how different expressions can be simplified or expanded, like in the binomial square example.Key features of algebraic expressions:

• Variables: These are usually letters (like \(x, y\)) representing unknown values.
• Terms: Parts of the expression separated by addition or subtraction, each term can be a constant, a variable, or a combination.
• Operations: The basic arithmetic operations that connect terms. These define the algebraic relationships between terms.

Grasping how algebraic expressions function and how they are manipulated is crucial for solving equations, working with functions, and entering advanced studies in mathematics.