Problem 101 - College Algebra

Question

Perform the indicated operations. $$[(7 x+5)+4 y][(7 x+5)-4 y]$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \((7x+5)^2 - (4y)^2\).

Step 1: Identify the form

There are two expressions of the form \((a + b)\) and \((a - b)\), where \(a = 7x + 5\) and \(b = 4y\). This is the form that fits difference of squares rule.

Step 2: Apply the difference of squares rule

According to the rule \([a^2 - b^2 = (a+b)(a-b)]\), we will now multiply the square of \(a\) with the square of \(b\).

Step 3: Calculate the squares

Calculate the square of \(a\) and \(b\) separately. The square of \(a\) will be \((7x+5)^2\) and the square of \(b\) will be \((4y)^2\).

Step 4: Subtract the squares

Subtract the square of \(b\) from the square of \(a\) to get the final result. So, the expression will be \((7x+5)^2 - (4y)^2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They do not have an equality sign, which differentiates them from equations. In the given exercise, the expression

• \( (7x+5) + 4y \quad \text{and} \quad (7x+5) - 4y \)

is an example of an algebraic expression.
Here, we see a combination of numbers, variables such as \(x\) and \(y\), and operations like addition and subtraction.
Algebraic expressions can be manipulated in various ways, such as simplification, evaluation, and applying certain rules like the difference of squares.
Understanding the structure of these expressions is essential in performing polynomial operations and finding simplified forms of these expressions. This foundational knowledge allows us to handle more complex algebraic problems later.

Squared Terms

A squared term arises when a base value is multiplied by itself. In algebra, you will often encounter situations that require you to find the square of a term. The exercise at hand utilizes the difference of squares formula.
For the expression

• \((7x+5)^2\)
• \((4y)^2\)

,we need to calculate each term's square. This means for \((7x+5)^2\), you will multiply \(7x+5\) by itself. Similarly, \((4y)^2\) involves multiplying \(4y\) by itself.
The principle of squaring ensures that regardless of whether the terms include variables or numbers, the method remains the same—each component is squared.
Knowing how to work with squared terms, especially in the context of polynomial operations, is crucial to applying rules like the difference of squares, as it helps simplify otherwise complex operations or expressions.

Polynomial Operations

Polynomial operations involve performing mathematical operations on polynomial expressions, like addition, subtraction, multiplication, or division.
In this exercise, the primary operation is applying the difference of squares formula, which is

• \(a^2 - b^2 = (a+b)(a-b)\)

.This particular rule is a powerful tool in algebra as it allows for simplifying expressions quickly. When evaluating the given problem

• \([(7x+5) + 4y][(7x+5) - 4y]\)

using polynomial operations, we identify \((7x+5)\) as the common term \(a\) and \(4y\) as \(b\).
This understanding lets us efficiently apply the difference of squares, simplifying a complex multiplication task into a simpler subtraction task:

• \((7x+5)^2 - (4y)^2\)

.These operations are crucial for students to streamline processes in mathematics and tackle advanced algebraic problems with confidence.