Problem 111 - Algebra and Trigonometry

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

Step 1: Identify the Given Terms

Firstly, identify the terms in the operation. From the exercise, the two terms can be considered as 'a' and 'b', \( a = (7x + 5)\) and \( b = 4y \). Where 'a' is represented by \( (7x + 5) \) and 'b' is represented by \( 4y \) in the difference of squares identity \( (a+b)(a-b) = a^2 - b^2 \).

Step 2: Apply the Algebraic Identity

Secondly, the identity to be applied is the difference of squares identity. Apply it as per the formula, \( a^2 - b^2 \), which means squaring both 'a' and 'b', and subtract 'b' squared from 'a' squared.

Step 3: Substitute and Simplify

Finally, substitute the values of 'a' and 'b' into the formula and simplify the expression further. So, \( a^2 = (7x + 5)^2 \) and \( b^2 = (4y)^2 \). Subtract \( b^2 \) from \( a^2 \) to get the final expression.

Key Concepts

Understanding Algebraic Identity

An algebraic identity is like a proven mathematical fact that holds true for all variables used in it. One of the most important identities in algebra is the difference of squares identity, which states that

• \( (a+b)(a-b) = a^2 - b^2 \)

This identity is very useful for simplifying expressions where you have a sum and a difference of the same two terms.
This is exactly what happens in our given problem where

• \( a = 7x + 5 \)
• \( b = 4y \)

By recognizing these terms, we can apply the difference of squares identity directly. This saves a lot of time and makes our calculations easier.

Techniques of Simplification

Simplification is a stepwise process of reducing an expression to its simplest form. It requires a good understanding of algebraic identities, like the difference of squares. In our problem, once the identity is applied, we need to compute and simplify

• \( a^2 = (7x + 5)^2 \)
• \( b^2 = (4y)^2 \)

Squaring these terms involves the expansion of the binomial square

• \[ (7x + 5)^2 = (7x)^2 + 2 \times 7x \times 5 + 5^2 = 49x^2 + 70x + 25 \]

Similarly,

• \( (4y)^2 = 16y^2 \)

Now, subtract the squared 'b' from the squared 'a'.
That means you calculate

• \( 49x^2 + 70x + 25 - 16y^2 \)

This step not only simplifies the expression but makes solving polynomial operations much faster.

Mastering Polynomial Operations

Polynomial operations are like playing with building blocks where each term represents a block. When conducting polynomial operations, you need to sum, subtract, multiply, or divide different polynomial terms. In our exercise, we start with two binomials within the brackets. These polynomials are made easier to handle using algebraic identities.
The exercise demonstrates how to use an identity to simplify and operate on these terms effectively. By applying the difference of squares, the expression is transformed into something manageable. This skill is fundamental in algebra as it helps in refining and solving polynomial equations efficiently. This approach is not only used in algebra but extends to calculus and beyond, wherever handling polynomials in a simplified manner becomes necessary. Knowing these operations makes you faster in solving complex problems and deepens your understanding of algebraic structures.