Problem 31 - Elementary and Intermediate Algebra

Question

Find each product. See Example 2. $$ \left(c+\frac{3}{4}\right)\left(c-\frac{3}{4}\right) $$

Step-by-Step Solution

Answer

The product is \(c^2 - \frac{9}{16}\).

Step 1: Identify the Structure

The given expression \((c + \frac{3}{4})(c - \frac{3}{4})\) fits the pattern of a difference of squares, which is \((a+b)(a-b) = a^2 - b^2\). In this case, \(a = c\) and \(b = \frac{3}{4}\).

Step 2: Apply the Difference of Squares Formula

Substitute \(a = c\) and \(b = \frac{3}{4}\) into the difference of squares formula \(a^2 - b^2\). This gives us \(c^2 - (\frac{3}{4})^2\).

Step 3: Calculate the Square of \(b\)

Calculate \((\frac{3}{4})^2\). This is done by squaring both the numerator and the denominator, giving us \(\frac{9}{16}\).

Step 4: Write the Final Expression

Now substitute \(\frac{9}{16}\) back into the expression, resulting in \(c^2 - \frac{9}{16}\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is used to express formulas and express relationships among numbers. In the context of this exercise, we use algebraic manipulation to simplify expressions and solve problems involving unknowns represented by variables like \(c\).

When working with algebra, you often need to identify patterns and apply specific formulas, like the difference of squares, to simplify expressions. The difference of squares formula is particularly useful because it transforms a multiplication problem into a subtraction problem, making calculations easier. By recognizing the pattern \((a+b)(a-b) = a^2 - b^2\), you can quickly find the product of two binomials that fit this structure.

This simplifies complex algebraic problems and aids in understanding relationships between variables.

Binomials

A binomial is an algebraic expression that contains exactly two terms. Binomials are commonly seen in algebra, and in the given problem, we have two binomials: \((c + \frac{3}{4})\) and \((c - \frac{3}{4})\). Each consists of a term involving the variable \(c\) and a constant.

Binomials can be added, subtracted, and multiplied with other polynomials. When multiplying two binomials, each term in the first binomial is multiplied by each term in the second binomial. However, the difference of squares formula simplifies this multiplication process greatly if the binomials are in the particular form \((a + b)(a - b)\).

Understanding binomials and their properties is key to mastering algebraic expressions and simplifying complex equations. This particular exercise showcased the elegant simplicity achieved when dealing with a specific type of binomial product.

Multiplication of Expressions

Multiplication of expressions involves finding the product of two or more expressions. When dealing with algebraic expressions, especially polynomials like binomials, this can often be lengthy if done manually term-by-term.

In this scenario, we recognize the expressions as a special case—the difference of squares. The difference of squares is a shortcut formula \((a + b)(a - b) = a^2 - b^2\) used to simplify such expressions. This pattern helps to avoid the long process of multiplying each term individually.

• Identify the form: Check if the multiplication fits the pattern \((a + b)(a - b)\).
• Apply the formula: Quickly simplify it to \(a^2 - b^2\).
• Execute the steps: Perform any calculations needed, such as squaring the constant \(b\).

Using multiplication rules such as these allows for faster and more efficient solving of algebraic problems, giving students the tools needed to tackle larger, more complex equations with ease.