Problem 81 - College Algebra

Question

Find each product. $$(7 x+3 y)(7 x-3 y)$$

Step-by-Step Solution

Answer

The product of the two binomials \( (7x + 3y)(7x - 3y) \) is \( 49x^2 - 9y^2 \).

Step 1: Identify the terms

Identify the terms in the binomials. Here, \(a\) is represented by \(7x\) and \(b\) is represented by \(3y\).

Step 2: Apply the difference of squares formula

Apply the formula for the difference of squares. Plugging \(7x\) for \(a\) and \(3y\) for \(b\) into the formula \(a^2 - b^2\), we get \((7x)^2 - (3y)^2\).

Step 3: Simplify the expression

Simplify the expression. Calculating the squares, we get \(49x^2 - 9y^2\).

Key Concepts

Difference of Squares

The difference of squares is a powerful algebraic concept used to simplify expressions involving binomials. It relies on the identity that states if you have a binomial multiplication of the form

• \( (a + b)(a - b) \)

then it simplifies to the form:

• \( a^2 - b^2 \).

This simplification works because when you expand the binomials using distributive property, the middle terms cancel each other out:

• \( (a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2 \).

In the given exercise,

• \( a = 7x \) and \( b = 3y \).

Thus,

• \( (7x + 3y)(7x - 3y) = (7x)^2 - (3y)^2 \).

This formula helps in quickly breaking down complex expressions into simpler ones without having to resort to lengthy multiplication processes.

Binomials

Binomials are algebraic expressions consisting of two distinct terms, usually joined by a plus or a minus sign. In our exercise example, we have two binomials

• \( 7x + 3y \) and \( 7x - 3y \).

These binomials represent combinations of variables and constants, often seen in various algebraic problems. Working with binomials involves applying different algebraic techniques like factorization, expansion, and simplification.
Binomials are foundational in creating more complex expressions called polynomials, where multiple terms are involved. Each term in a binomial can itself be a product of constants and variables, providing a basis for further manipulation and problem-solving. When multiplying binomials, recognizing structures like the difference of squares enables straightforward simplification. Understanding how these two terms interact in polynomial operations is key to mastering algebraic expressions.

Polynomial Multiplication

Polynomial multiplication refers to the process of multiplying two polynomials together, which involves multiplying every term in one polynomial by every term in the other polynomial. In simpler terms, it's about distributing all terms to form new terms.
In the context of our problem

• \( (7x + 3y) \text{ and } (7x - 3y) \),

we leverage a shortcut through the difference of squares,

• \( (a + b)(a - b) = a^2 - b^2 \).

This shortcut avoids the repetitive expansion:

• \( (7x \cdot 7x) + (7x \cdot -3y) + (3y \cdot 7x) + (3y \cdot -3y) \)

Instead, recognizing the structure as a difference of squares directly gives us:

• \( (7x)^2 - (3y)^2 \)

Thus, the polynomial multiplication is simplified to just finding the square of each binomial term without the intermediate steps.