Problem 111 - College Algebra Essentials

Question

In Exercises 111–113, perform the indicated operations. $$[(7 x+5)+4 y][(7 x+5)-4 y]$$

Step-by-Step Solution

Answer

The simplified version of the given expressions is: \( 14x + 10 \)

Step 1: Identify the binomial expressions

In this problem, \( (7x + 5 + 4y) \) and \( (7x + 5 - 4y) \) are the binomial expressions.

Step 2: Apply the difference of squares formula

The difference of squares formula is given by \( a^2 - b^2 = (a + b)(a - b) \). In this context, \( a = (7x + 5 + 4y) \) and \( b = (7x + 5 - 4y) \), implies that \( a - b = 0 \) and \( a + b = 2(7x + 5) \).

Step 3: Simplify the expression

Simplify the expression via distributing the '2' within the brackets, this gives us \( 2*7x + 2*5 = 14x + 10 \).

Key Concepts

Binomial Expressions

Binomial expressions are algebraic expressions that contain exactly two terms.
They are often connected by a plus (+) or minus (-) sign. In the exercise given, we have two binomial expressions:

• \((7x + 5 + 4y)\)
• \((7x + 5 - 4y)\)

Understanding binomial expressions is crucial when dealing with algebraic operations like multiplication, as it forms the building blocks for more complex expressions.
Binomials are the fundamental pieces that later combine to form bigger algebraic structures. Being able to identify and manipulate these expressions helps build a strong foundation in algebra.

Difference of Squares

The difference of squares is a special algebraic identity that simplifies the product of two conjugate binomials.
It is represented by the formula: \(a^2 - b^2 = (a + b)(a - b)\).
In our problem, the expression is naturally set up to use this formula.

• The structure \([(7x + 5) + 4y][(7x + 5) - 4y]\) fits exactly the difference of squares pattern.
• This concept enables us to simplify the expression efficiently without performing direct multiplication.

By identifying the components \((a)\) and \((b)\), we reduce complex multiplication to straightforward subtraction.
This results in an elegant simplification.

Simplification

Simplification involves rewriting an expression in a more concise or easily interpretable form.
In this exercise, the application of the difference of squares formula allows the expression to be condensed from its original form. By observing \((a + b)(a - b)\) as part of the difference of squares formula, it reduces

• to a simpler mathematical form: \(a^2 - b^2\).
• This further simplifies down to \((14x + 10)\) using distributive property.

Simplifying expressions helps to make calculations easier and often reveals additional properties or solutions that were not visible before.
This process is key in algebra for revealing simpler truths from more complex expressions.

Distributive Property

The distributive property is a fundamental principle in arithmetic and algebra, crucial for simplifying expressions.
It states that multiplying a single term by terms inside a parenthesis can be done individually to each term. Mathematically, this is expressed as \(a(b + c) = ab + ac\).
In the context of our exercise, the simplification step emerges from distributing:

• Here, distributing "2" gives \(14x + 10\).
• This property breaks down a potentially complex multiplication into simple, manageable terms.

The distributive property is valuable in simplification and essential in expanding and reducing expressions in algebra.
Mastering it aids in manipulating and simplifying a wide range of algebraic problems.