Problem 46 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (x-4)^{2} $$

Step-by-Step Solution

Answer

\(x^2 - 8x + 16\)

Step 1: Identify the Binomial Terms

In this problem, the binomial is \((x-4)\) and it is being squared. In the algebraic principle, this corresponds to \(a = x\) and \(b = 4\).

Step 2: Apply the Algebraic Principle

Now substitute these values into the principle \((a-b)^2 = a^2 - 2ab + b^2\). This gives us \((x - 4)^2 = x^2 - 2*x*4 + 4^2\).

Step 3: Simplify Further

This simplifies to \(x^2 - 8x + 16\).

Step 4: Write Down the Final Answer

The final result of \((x-4)^2\) is \(x^2 - 8x + 16\) after completing the simplification.

Key Concepts

Binomial Expansion

Binomial expansion is a fundamental concept in algebra where we take a binomial, a sum or difference of two terms, and expand it into a polynomial. When expanding a binomial, it often involves squaring or raising it to a higher power. The formula for expanding a squared binomial is given by:

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)

These formulas come in handy whenever you need to square a binomial expression like \((x-4)^2\). Recognizing the structure and using these templates helps solve problems quickly and accurately. By substituting the identified terms into the appropriate format, we can easily perform the calculations and reach the expanded polynomial.

Squared Binomials

A squared binomial refers to an expression where a binomial term is raised to the power of two. In essence, you’re multiplying the binomial by itself. Let’s consider the binomial \((x-4)^2\). To square it, you use the format \((a-b)^2 = a^2 - 2ab + b^2\), where in this case, *a* represents *x* and *b* is *4*.

Squaring of this binomial consists of three key steps:

• Calculate \(a^2\): Here, \(x^2\) is the result when *x* is squared.
• Calculate \(-2ab\): Multiply *x* by *4* and by *-2*, resulting in \(-8x\).
• Calculate \(b^2\): Multiply 4 by itself to get \(16\).

Finally, by summing the results from the individual components, we derive the resulting expression \(x^2 - 8x + 16\). This complete item represents the squared binomial.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to its simplest form. This process requires combining like terms and performing arithmetic operations to streamline the expression. In the problem of squaring \((x-4)\), after applying the binomial expansion, we arrive at an un-simplified polynomial: \(x^2 - 2 \cdot x \cdot 4 + 4^2\).

Here's how simplification works:

• Calculate each term separately: \(x^2\), \(-8x\), and \(+16\).
• Check if there are any like terms. Here, each term is distinct: \(x^2\), \(x\), and constant term are all unique.
• Combine all the terms into a final result: \(x^2 - 8x + 16\). This is the simplest form for the expanded binomial.

Simplifying expressions helps not only in obtaining the correct answer but also plays a crucial role in solving greater algebraic equations effectively.