Problem 28 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(x-5)(x+5)(x-8)$$

Step-by-Step Solution

Answer

The product is \(x^3 - 8x^2 - 25x + 200\).

Step 1: Identify the Special Pattern

The expression \((x-5)(x+5)\)involves the sum and difference of the same two terms. This is a special pattern known as the difference of squares, which states that \((a-b)(a+b) = a^2 - b^2\). Recognize that the first part of the product uses this pattern.

Step 2: Apply the Difference of Squares Formula

Apply the difference of squares formula to \((x-5)(x+5)\):\[ x^2 - 5^2 = x^2 - 25 \]

Step 3: Substitute Back into the Original Expression

Substitute the result from Step 2 into the original expression:\[(x^2 - 25)(x - 8)\].

Step 4: Distribute to Expand the Expression

Now, you need to distribute \((x^2 - 25)\) into \((x - 8)\):First, distribute \(x^2\):\[x^2(x - 8) = x^3 - 8x^2\]Then, distribute \(-25\):\[-25(x - 8) = -25x + 200\]Combine these products to expand the expression.

Step 5: Combine Like Terms

After distributing, combine the terms: \[x^3 - 8x^2 - 25x + 200\].

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols such as addition, subtraction, multiplication, and division. They are used to represent quantities and relationships in algebra. An example, like

• \( (x-5)(x+5)(x-8) \),

features binomials and shows the use of variables within parentheses. Understanding algebraic expressions involves knowing how to manipulate these symbols to simplify or solve equations.

When dealing with algebraic expressions, operations such as addition, subtraction, and multiplication are commonly used to combine or alter them. This is especially important when simplifying expressions or solving equations. Recognizing special patterns, such as the multiplication of binomials, simplifies the process of dealing with these expressions. By mastering these rules, you can handle complex mathematical problems more efficiently.

Difference of Squares

The difference of squares is a special algebraic pattern that simplifies certain types of binomial multiplication. When you encounter an expression like

• \((a-b)(a+b)\),

this is recognized as a difference of squares because it results in

• \(a^2 - b^2\).

This pattern provides a shortcut, making calculations faster by avoiding the need for separate binomial multiplications.

For example, in the expression \((x-5)(x+5)\), the terms involve a subtraction and addition of the same number, 5. Applying the difference of squares pattern immediately simplifies it to \(x^2 - 25\). This reduction not only saves time but also helps avoid potential arithmetic errors, ensuring calculations are accurate and efficient.
Understanding this concept is crucial, as it helps solve problems with binomials more effectively and can often be applied in various algebraic contexts to simplify or factor expressions.

Polynomial Expansion

Polynomial expansion is the process of multiplying polynomials to express them as a sum of terms. In the exercise

• \( (x^2 - 25)(x - 8) \),

this involves distributing each term of one polynomial into every term of the other.

The steps include:

• First distributing \(x^2\) across the binomial \(x - 8\),
• resulting in \(x^3 - 8x^2\).
• Then, distributing \(-25\) across \(x - 8\),
• resulting in \(-25x + 200\).
• Finally, you combine the terms \(x^3 - 8x^2 - 25x + 200\) to complete the polynomial expansion.

This method is foundational to algebra as it allows you to represent a complex expression as a sum of simpler terms. It lays the groundwork for solving higher-level polynomial equations, understanding their roots, and applying them to real-world scenarios. Mastering polynomial expansion opens the door to deeper algebraic concepts and operations.