Problem 48 - Algebra A Combined Function

Question

Factor each completely. $$ (3 x+y)^{2}-25 $$

Step-by-Step Solution

Answer

The expression factors to \((3x + y - 5)(3x + y + 5)\).

Step 1: Recognize the structure

The expression \((3x + y)^2 - 25\) follows the form of a difference of squares, which is \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = 3x + y\) and \(b = 5\) because \(5^2 = 25\).

Step 2: Apply the difference of squares formula

Using the difference of squares formula \((a-b)(a+b)\), we replace \(a\) and \(b\) with \(3x + y\) and \(5\) respectively, resulting in \((3x + y - 5)(3x + y + 5)\).

Step 3: Write the final factored expression

Thus, the factored form of \((3x + y)^2 - 25\) is \((3x + y - 5)(3x + y + 5)\).

Key Concepts

Difference of Squares

The difference of squares is a mathematical concept used to factor certain types of quadratic expressions. It's a special case when you have a subtraction, or difference, between two square terms. In mathematical form, it is expressed as:\[a^2 - b^2 = (a - b)(a + b)\]This formula states that any expression structured as a square number minus another square number can be factored into a product of two binomials. Each binomial contains the original terms with opposite signs.In the given exercise, the expression \((3x + y)^2 - 25\) perfectly fits the difference of squares schema:

• \(a^2\) corresponds to \((3x + y)^2\)
• \(b^2\) corresponds to 25, which is \(5^2\)

By identifying that this is a difference of squares, you can easily factor the expression using the formula.

Factoring Expressions

Factoring expressions is an essential skill in algebra, allowing you to simplify expressions or solve equations. Essentially, factoring involves breaking down a complex expression into simpler components or 'factors' that, when multiplied together, give back the original expression.In the example \((3x + y)^2 - 25\), it is expressed initially as a difference of two squares. Recognizing this form allows you to apply a specific factorizational technique tailored for such expressions. Here, we identified:

• \(a = 3x + y\), hence \(a^2 = (3x + y)^2\)
• \(b = 5\), with \(b^2 = 25\)

Applying the difference of squares formula, we can break down the expression into two factors:\[(3x + y - 5)(3x + y + 5)\]Factoring expressions like this helps in simplifying problems and is also used when solving equations by setting each factor equal to zero to find possible solutions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations such as addition, subtraction, multiplication, and division. They represent mathematical relationships and are the foundational elements of algebra.The specific algebraic expression in this exercise is \((3x + y)^2 - 25\). This expression includes:

• A binomial \((3x + y)\)
• An exponent indicating a square
• A subtraction operation signifying the difference of squares

Working with such expressions requires understanding how to manipulate them. This allows for simplifying or transforming them into usable forms, such as factoring.In tackling algebraic expressions, it is crucial to:

• Recognize patterns, like the difference of squares
• Use appropriate formulas and techniques for simplification and solution

Algebraic expressions are everywhere in mathematics and mastering them is key to solving more advanced problems effectively.