Problem 105 - Algebra A Combined Function

Question

Perform each indicated operation. $$ (x-4)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(x^2 - 8x + 16\).

Step 1: Identify the Formula

The expression \( (x-4)^{2} \) is a square of a binomial. We need to expand it using the formula for squaring a binomial: \[ (a - b)^2 = a^2 - 2ab + b^2 \].

Step 2: Assign Values for Squaring

Identify \(a\) and \(b\) in the expression \( (x-4)^2 \). Here, \( a = x \) and \( b = 4 \).

Step 3: Apply the Formula

Substitute \(a\) and \(b\) into the formula: \[ a^2 - 2ab + b^2 = x^2 - 2(x)(4) + 4^2 \].

Step 4: Simplify the Expression

Calculate each term: \( x^2 \) remains \( x^2 \), \(-2(x)(4)\) becomes \(-8x\), and \(4^2\) is \(16\). So, the expression simplifies to \[ x^2 - 8x + 16 \].

Key Concepts

Binomial Expansion

Binomial expansion is a process where we multiply out expressions that are raised to a power, and they involve two terms (hence "bi"). The given problem, \((x-4)^{2}\), is a classic example of this. To expand a binomial squared, we apply a special formula, commonly known as the binomial theorem for squares: \((a-b)^2 = a^2 - 2ab + b^2\). This formula allows us to efficiently expand the expression without having to multiply \((x-4)(x-4)\) step by step.

• First, identify the two terms of the binomial; in this case, \(a = x\) and \(b = 4\).
• Next, apply the binomial square formula directly, where you square the first term, multiply the two terms and double them, and then square the second term.

By using this approach, we avoid lengthy expansions and streamline the calculation process. Binomial expansion is a valuable skill, especially as you move into more advanced algebra, where expression manipulation becomes more complex.

Polynomial Simplification

Simplifying polynomials is all about making an algebraic expression neater and more manageable. In our example, after expanding \((x-4)^2\), we need to simplify the result: \(x^2 - 8x + 16\). This means we combine like terms and ensure that no further simplification is possible.

• First, check each term is in its simplest form: \(x^2\), \(-8x\), and \(16\) do not have like terms to combine with.
• It's crucial to verify each operation is done correctly, such as ensuring coefficients are multiplied accurately and signs are handled properly.

The goal is a clean, simplified polynomial that can be efficiently used in further operations such as solving equations or graphing. Proper simplification lays the groundwork for continued mathematical manipulation.

Square of a Binomial

The square of a binomial involves taking a two-term algebraic expression and multiplying it by itself. The standard formula for squaring a binomial is \((a-b)^2 = a^2 - 2ab + b^2\). This formula is key to solving the exercise provided.

• The terms \(a\) and \(b\) are identified from the binomial \((x-4)\), where \(a = x\) and \(b = 4\).
• We then use the formula to efficiently expand the expression, squaring \(x\) to get \(x^2\), computing \(-2(x)(4)\) to get \(-8x\), and finally \((4)^2 = 16\).

Understanding the square of a binomial is a fundamental skill in algebra. It simplifies calculations and is an essential tool in solving more complex polynomial equations later in math studies. By mastering this technique, you'll handle algebraic expressions with increased ease and precision.