Problem 76 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (3 x+5)(3 x-5) $$

Step-by-Step Solution

Answer

\((3x+5)(3x-5) = 9x^2 - 25\).

Step 1: Identify the Pattern

Notice that the given expression, \((3x + 5)(3x - 5)\), is a difference of squares, which follows the pattern \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 3x\) and \(b = 5\).

Step 2: Square Each Term

Using the difference of squares formula, we square each term: \(a^2 = (3x)^2 = 9x^2\) and \(b^2 = 5^2 = 25\).

Step 3: Subtract the Squares

Plug the squared terms into the formula: \(a^2 - b^2 = 9x^2 - 25\). So, \((3x + 5)(3x - 5) = 9x^2 - 25\).

Step 4: Simplify

The expression \(9x^2 - 25\) is already in its simplest form as a difference of squares.

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra and mathematics in general. These expressions combine variables, numbers, and operations like addition, subtraction, multiplication, and division. In our exercise, the algebraic expression

• consists of terms with variables and coefficients, such as \(3x\).
• Each part of the expression has a specific role. For example, the coefficient \(3\) in \(3x\) indicates how many times the variable \(x\) is being used.

Understanding algebraic expressions involves learning how to manipulate these variables and constants effectively. This means mastering operations and transformations on these expressions, such as expansion and factoring. In exercises involving a difference of squares, recognizing patterns allows you to simplify complex expressions.
Using known formulas leads to easier manipulation and interpretation of algebraic expressions. This is essential for progressing in algebra and applying these skills to solve real-world problems.

Factoring

Factoring in algebra is the process of breaking down an expression into its simpler components or "factors." This technique is incredibly useful, especially when dealing with polynomials, as it allows you to simplify and solve equations more efficiently. In the context of the difference of squares, factoring identifies how two terms multiply to create a specific expression.
For example,

• when we factor the expression \((3x + 5)(3x - 5)\), we identify it as a difference of squares.
• This means that the factors are structured in a particular way: one term is added, and the other is subtracted.

The formula \((a + b)(a - b) = a^2 - b^2\) clearly shows this relationship. Expanding and simplifying the factors gives us the product \(9x^2 - 25\).
Factoring is not just a method but an essential skill in algebra that simplifies complex problems into understandable solutions. Mastery over factoring techniques, including recognizing a difference of squares, is crucial for solving higher-level algebraic problems.

Polynomials

Polynomials are expressions made up of variables raised to integer powers and multiplied by coefficients. They can have one or more terms, such as a single variable or a combination of different powers and roots. A polynomial's degree is determined by the highest power of the variable.

• In our exercise, \(9x^2 - 25\) is recognized as a difference of squares and is part of a polynomial expression.
• The squared term \(9x^2\) has a higher degree than the constant \(25\), making it the leading term of the polynomial.

Understanding polynomials involves operations like addition, subtraction, multiplication, and factoring. Recognizing patterns, such as the difference of squares, allows for simplifying and solving polynomial equations.
This involves writing polynomials in standard form or breaking them into simpler factors. Besides calculating, polynomials model various real-world situations, positioning them as crucial concepts in both mathematics and scientific disciplines. Becoming proficient in manipulating polynomials is valuable for mathematical fluency and problem-solving.