Problem 32 - Elementary and Intermediate Algebra

Question

Find each product. See Example 2. $$ \left(m+\frac{4}{5}\right)\left(m-\frac{4}{5}\right) $$

Step-by-Step Solution

Answer

The product is \(m^2 - \frac{16}{25}\).

Step 1 - Recognize the Formula

Identify that the expression \((m + \frac{4}{5})(m - \frac{4}{5})\) fits the pattern of a difference of squares formula: \((a + b)(a - b) = a^2 - b^2\).

Step 2 - Identify Terms

In the formula \((a + b)(a - b) = a^2 - b^2\), recognize that \(a = m\) and \(b = \frac{4}{5}\).

Step 3 - Apply the Difference of Squares Formula

Substitute the identified terms into the difference of squares formula: \(a^2 - b^2 = m^2 - \left(\frac{4}{5}\right)^2\).

Step 4 - Calculate Squares

Calculate the square of each term: \(m^2 = m^2\) and \(\left(\frac{4}{5}\right)^2 = \frac{16}{25}\).

Step 5 - Substitute and Simplify

Substitute back into the expression: \(m^2 - \left(\frac{16}{25}\right) = m^2 - \frac{16}{25}\).

Key Concepts

Understanding Polynomials

A polynomial is a type of algebraic expression that involves sums and/or differences between terms, which are made up of variables raised to non-negative integer powers and multiplied by coefficients.

• Each term in a polynomial consists of a coefficient—a numerical factor—and a variable with an exponent.
• Polynomials are often classified by their degree. The degree is determined by the term with the largest sum of exponents on its variables.
• For example, in the expression \(3x^2 + 5x + 7\), \(x^2\) is a term where the degree is 2, 5 is the coefficient of \(x\), and 7 is a constant term.

Polynomial expressions can become more complex as the number of terms increases, but they're very useful in representing a wide array of mathematical relationships and functions. They are foundational in algebra and often appear in algebraic equations, factorization, and various forms of graphing.

Exploring Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations that represent a value or set of values. They are the building blocks of algebra and consist of terms, where each term is a product of a number (coefficient) and a variable raised to an exponent.

• Expressions do not always have an "equals" sign; if they do, they turn into equations.
• For instance, \(2x + 3\) is an algebraic expression meaning twice a number \(x\) plus three.
• A difference of squares, like in the problem \((m + \frac{4}{5})(m - \frac{4}{5})\), is a special type of algebraic expression that can be easily factored, using the formula \(a^2 - b^2\).

Algebraic expressions enable us to solve problems by applying mathematical operations to reveal relationships between different mathematical quantities. They provide a concise way to represent complex situations and their possible solutions inform much of the work done in algebra.

Factoring Simplified

Factoring is the process of breaking down a complex expression into simpler factors or terms that, when multiplied together, return to the original expression. It’s a fundamental concept in algebra useful for simplifying expressions and solving equations.

• One of the most common patterns is the difference of squares: \((a + b)(a - b) = a^2 - b^2\). It allows quick factorization of expressions like \(m^2 - \frac{16}{25}\).
• In our problem, we used the difference of squares to simplify \((m + \frac{4}{5})(m - \frac{4}{5})\) easily into \(m^2 - \frac{16}{25}\).
• Factoring makes solving algebraic equations easier by reducing each part into its simplest components, revealing roots or solutions.

Factoring not only simplifies expressions but also prepares them for further operations or integration into larger equations. Mastering factoring techniques accelerates solving complex algebraic problems efficiently.