Problem 82 - College Algebra Essentials

Question

In Exercises 67–82, 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 of the given polynomials is \(9x^2 y^4 - 16y^2\).

Step 1: Identify the Terms

The given expression is \((3xy^2 - 4y)(3xy^2 + 4y)\). Here, the first term in each bracket is \(3xy^2\) and the second term is \(-4y\) and \(4y\), respectively.

Step 2: Apply the Distributive Property (FOIL)

FOIL stands for 'First, Outer, Inner, Last', meaning we multiply the terms in this order and sum them up. First term: \(3x y^2 * 3x y^2 = 9x^2 y^4\), Outer terms: \(3x y^2 * 4y = 12x y^3\), Inner terms: \(-4y * 3x y^2 = -12x y^3\), Last terms: \(-4y * 4y = -16y^2\).

Step 3: Simplify the Resulting Expression

Add all terms together: \(9x^2 y^4 + 12x y^3 - 12x y^3 - 16y^2\). Note that the outer and inner terms cancel out, which results in \(9x^2 y^4 - 16y^2\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental in algebra. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables are symbols that represent unknown values and can take on various numerical values. For example, in the expression \(3xy^2 - 4y\), \(x\) and \(y\) are variables, while \(3\) and \(4\) are coefficients that quantify the variables, and the minus sign (\