Problem 46 - College Algebra

Question

Find each product. $$(x-4)^{2}$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Problem to Visualize the Multiplication

Begin by rewriting the problem as \((x-4) * (x-4)\) to make it clear that there is a multiplication of two binomials.

Step 2: Apply Distributive Property

Now use the FOIL method to multiply. Multiply the first terms in each binomial together to get \(x * x = x^{2}\). Multiply the outer terms together to get \(x * -4 = -4x\). Multiply the inner terms together to get \(-4 * x = -4x\). Lastly, multiply the last terms in each binomial together to get \(-4 * -4 = 16\).

Step 3: Simplify the Expression

Now, add the multiplication terms together. This simplifies to \(x^{2} - 4x - 4x + 16\). When combining like terms, you will get \(x^{2} - 8x + 16\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to distribute a single term across terms inside parentheses. For multiplying binomials, the distributive property is often applied in a structured way like the FOIL method.

When you have an expression like \(a(b + c)\), you use the distributive property to expand it:

• Multiply \(a\) by \(b\) to get \(ab\).
• Multiply \(a\) by \(c\) to get \(ac\).

This results in the expanded expression \(ab + ac\).

For more complex expressions, especially those involving binomials like \(x - 4\), you apply the distributive property through each term within the parentheses. This is precisely what simplifies and systematizes the multiplication process, ensuring nothing is missed.

FOIL Method

The FOIL method is a nifty technique for multiplying two binomials. It stands for First, Outer, Inner, and Last, representing the pairs of terms you multiply together.

Here's how it works for \( (x-4)(x-4) \):

• First: Multiply the first terms, \(x * x\), giving you \(x^2\).
• Outer: Multiply the outer terms, \(x * -4\), resulting in \(-4x\).
• Inner: Multiply the inner terms, \(-4 * x\), which gives \(-4x\).
• Last: Multiply the last terms, \(-4 * -4\), providing the result \(+16\).

By following FOIL, you systematically ensure each term contributes to the product. Combined, these terms eventually simplify to show the completed expression. Remember, practice makes perfect!

Simplification of Expressions

Simplifying expressions is about combining like terms to make an expression easier to understand. After multiplying, as seen with \( (x-4)(x-4) \), you'll end up with terms like \( x^2 - 4x - 4x + 16 \).

To simplify:

• Combine like terms. Here, \(-4x\) and \(-4x\) are like terms since they both involve \(x\).
• Adding them gives you \(-8x\).
• The expression then becomes \(x^2 - 8x + 16\).

Simplification is crucial for solving equations and understanding expressions fully. It helps you see the clear and final form of an equation, making calculations and interpretations more straightforward. Keep practicing to develop intuition over which terms can combine, always leading to clearer expressions.