Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\(\left(3 x^{2}-4 y^{2}\right)\left(3 x^{2}+4 y^{2}\right)\)

Step-by-Step Solution

Answer

The product of \((3 x^{2}-4 y^{2})\) and \((3 x^{2}+4 y^{2})\) is \(9x^4 - 16y^4\).

Step 1: Understand the basic concept

The difference of squares method states that \(a^2 - b^2 = (a+b)(a-b)\). In this case, \(a = 3x^2\) and \(b = 4y^2\). The given binomials \((3x^2 - 4y^2)\) and \((3x^2 + 4y^2)\) are in the form of \((a - b)\) and \((a + b)\) respectively.

Step 2: Apply the formula

Now, substitute \(a\) and \(b\) in the formula \(a^2 - b^2 = (a+b)(a-b)\). Here, \(a+b = (3x^2 + 4y^2)\) and \(a-b = (3x^2 - 4y^2)\). Hence, the product is \(a^2 - b^2 = (3x^2)^2 - (4y^2)^2\).

Step 3: Calculate the product

Calculate the square of \(3x^2\) and \(4y^2\) to get \(9x^4\) and \(16y^4\) respectively. So, \(a^2 - b^2 = 9x^4 - 16y^4\).

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It allows us to express relationships and patterns in a general form. In algebra, letters or symbols are used to represent numbers. This makes it easier to write and manipulate mathematical expressions.
A crucial aspect of algebra is understanding how to work with equations and expressions. By using algebraic methods, we can solve for unknown quantities and find out their relationships with others.
An important skill in algebra is recognizing patterns, such as the difference of squares. This pattern helps in simplifying expressions and solving equations efficiently.

Exploring Binomials

Binomials are algebraic expressions that consist of two terms separated by either a plus or a minus sign. Examples of binomials include

• \((a + b)\)
• \((x - y)\)

Binomials can often be combined through operations such as addition, subtraction, multiplication, or division.
A particularly useful technique when multiplying binomials is the difference of squares. This arises when we have two binomials of the form

• \((a + b)\)
• \((a - b)\)

This results in \(a^2 - b^2\), simplifying the multiplication process considerably.

Understanding Polynomials

Polynomials are algebraic expressions that consist of multiple terms. These terms can include variables raised to whole number powers and constants. A simple example of a polynomial is

• \(3x^2 - 4y^2\)
• \(5x^3 + x - 6\)

Polynomials are often classified based on the number of terms they contain. A polynomial with two terms is specifically known as a binomial.
When working with polynomials, it's important to understand operations such as addition, subtraction, multiplication, and sometimes division. This involves combining like terms and using techniques such as the distributive property.
In our exercise, the difference of squares is a method that simplifies the multiplication of two binomials, resulting in a polynomial expression. The product of the given binomials is another polynomial, deduced using this method.