Problem 20 - 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. $$(t+8)(t-8)$$

Step-by-Step Solution

Answer

The product is \(t^2 - 64\).

Step 1: Identify the Problem Type

The exercise requires finding the product \((t+8)(t-8)\). Notice that this is a difference of squares pattern, \((a+b)(a-b) = a^2 - b^2\). In our case, substitute \(t\) for \(a\) and \(8\) for \(b\).

Step 2: Apply the Difference of Squares Formula

Using the difference of squares pattern, the expression \((t+8)(t-8)\) is equivalent to \(t^2 - 8^2\). This simplifies the multiplication process by applying this well-known pattern.

Step 3: Compute the Squares

Now, compute the squares of the individual terms: \(t^2\) remains \(t^2\), and \(8^2\) is \(64\).

Step 4: Write the Final Expression

Replace the squares with their computed values in the expression \(t^2 - 8^2\) to obtain \(t^2 - 64\). This is the simplified product of the original binomials.

Key Concepts

Multiplying Binomials

When faced with the task of multiplying binomials, one effective strategy is to use special patterns that allow for quick and efficient computation. In the given exercise, you encounter the binomials \((t+8)\) and \((t-8)\). The goal is to calculate their product. By recognizing that this expression fits the difference of squares pattern, you can apply a useful formula instead of expanding and combining terms the conventional way.

• Difference of squares formula: \((a+b)(a-b) = a^2 - b^2\)

This pattern is a shortcut for multiplying binomials that have terms differing only in sign. Recognizing such patterns can simplify calculations and reduce errors.

Algebraic Expressions

Algebraic expressions, such as \((t+8)(t-8)\), consist of variables, numbers, and arithmetic operations. They sometimes require simplification to reveal their essence. To address them effectively, understanding operations like addition, subtraction, multiplication, and division is vital.

• Variables represent unknown quantities (e.g., \(t\)).
• Arithmetic operations manipulate these variables and constants within expressions.

The task of simplifying involves transforming expressions like our binomials into a more comprehensible form, which might involve recognizing patterns such as the difference of squares.

Special Products

Special products refer to algebraic products that result from multiplying certain binomials in a way that reflects specific recognizable patterns. In our case, \((t+8)(t-8)\) exemplifies a special product known as a difference of squares.

• The formula for difference of squares: \(a^2 - b^2\).
• This formula can be quickly applied to simplify products without direct term-by-term multiplication.

These special patterns are valuable tools in your mathematical toolkit for simplifying complex algebraic expressions quickly and accurately. When employed correctly, they save computational time and enhance understanding of algebraic structures.