Problem 82 - College Algebra

Question

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

Step-by-Step Solution

Answer

The product \((3 x y^{2}-4 y)(3 x y^{2}+4 y)\) equals to \(9x^{2}y^{4} - 16y^{2}\).

Step 1: Apply the distributive property

Apply the distributive property, starting by multiplying the first terms (3xy^2)(3xy^2), then the outside terms (3xy^2)(4y), then the inside terms (-4y)(3xy^2), and finally the last terms (-4y)(4y).

Step 2: Perform the multiplication

Perform the multiplication for each term. (3xy^2)(3xy^2) leads to \(9x^{2}y^{4}\), (3xy^2)(4y) leads to \(12x y^{3}\), (-4y)(3xy^2) leads to \(-12x y^{3}\), and (-4y)(4y) leads to \(-16y^{2}\).

Step 3: Combine like terms

After performing the multiplication, combine like terms if any. However, in this case, there are no like terms.

Step 4: Write down the product

The product is the sum of all the terms. We get the product \((3 x y^{2}-4 y)(3 x y^{2}+4 y) = 9x^{2}y^{4} - 16y^{2}\), where \(12x y^{3}\) and \(-12x y^{3}\) cancel each other out.

Key Concepts

Distributive Property

The distributive property is a cornerstone in algebra that allows us to multiply a single term by two or more terms within a parenthesis. It is often used in polynomial multiplication, like in our exercise with \( (3xy^2 - 4y) (3xy^2 + 4y) \). The process involves taking each term from the first polynomial and multiplying it by every term of the second polynomial.

To visualize: \((a + b)(c + d) = ac + ad + bc + bd\). This pattern is what you use no matter how complex the expression is. In the exercise, we apply the property step by step. Although it might seem tedious at first, especially with polynomials of higher degrees, mastering this concept will make subsequent algebra tasks much smoother.

Combining Like Terms

When we talk about combining like terms, we refer to the process of simplifying an algebraic expression by adding or subtracting terms that have the same variables raised to the same power. It’s like gathering apples with apples and oranges with oranges. In the context of our example, \((3 x y^{2})(4 y)\) and \((-4y)(3xy^2)\) may look different, but because they have the variables 'x' and 'y' to the same powers, these can be considered like terms.

However, in this particular instance, when we perform the multiplication, those terms cancel each other out because they are equal in magnitude but opposite in sign, resulting in zero. This shows the importance of carefully identifying like terms -- they can sometimes eliminate each other, simplifying your expression.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and sometimes division. In this exercise, we focus on multiplication, which requires us to use both the distributive property and combine like terms. It’s important to orderly execute each operation step by step, as skipping a step can lead to errors.

For efficient multiplication of polynomials, the following method usually works best: multiply each term in the first polynomial by each term in the second polynomial, write the result in a structured way, and then combine like terms. Understanding the underlying structure of polynomials is key to success with these operations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In our exercise, \(3xy^2 - 4y\) and \(3xy^2 + 4y\) are both algebraic expressions consisting of variables (x and y) and numbers (3, 4). These expressions can hold immense power as they provide the means to describe relationships, patterns, and general mathematical truths.

Remember that each part of an algebraic expression has a name: the number before the variable (like 3 in 3xy^2) is called a coefficient, the variables (x, y) are sometimes referred to as unknowns or literals, and the numbers without variables (like 4 in 4y) are constants. When dealing with algebraic expressions, especially in polynomial multiplication, keeping track of these components is critical to managing the operations correctly.