Problem 81 - College Algebra Essentials

Question

In Exercises 67–82, find each product. $$\left(7 x y^{2}-10 y\right)\left(7 x y^{2}+10 y\right)$$

Step-by-Step Solution

Answer

Therefore, the product of \( (7xy^2-10y) \) and \( (7xy^2+10y) \) is \( 49x^2y^4 - 100y^2 \)

Step 1: Identify the Form

This is an algebraic expression comprising of two binomials, of the form \( (a-b)(a+b)\). Here, \( a = 7xy^2 \) and \( b = 10y\). We will use this form to simplify the equation.

Step 2: Apply Difference of Squares Rule

Using the difference of squares rule, which states that \( (a-b)(a+b) = a^2 - b^2 \), we can simplify the expression. This rule is applied here because it simplifies binomial multiplication when the expressions have matching terms, one with a plus and the other with a minus.

Step 3: Compute the Solution

On substitution in \( a^2 - b^2 \), \( a=7xy^2\) and \( b=10y \), our expression simplifies to \( (7xy^2)^2 - (10y)^2 = 49x^2y^4 - 100y^2 \)

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. For instance, in the exercise \( (7 x y^2-10 y)(7 x y^2+10 y) \), the variables are \( x \) and \( y \), while numbers and operations include 7, 10, addition, subtraction, and multiplication. Each expression within the parentheses is a binomial since it consists of two terms.

To simplify algebraic expressions, especially products like in the exercise, we rely on rules and patterns such as the difference of squares. Recognizing these patterns makes algebra more approachable and solving equations less daunting. Understanding how to combine and simplify like terms, expand products, and factor are also crucial skills in working with algebraic expressions.

Binomial Multiplication

Binomial multiplication involves finding the product of two binomials, which are expressions containing two terms. For example, \( (7 x y^2) \) and \( (-10 y) \) form one binomial, while \( (7 x y^2) \) and \( (+10 y) \) form the second. When multiplying binomials, we typically apply the FOIL method - multiplying first, outer, inner, and last terms.

However, when dealing with special cases like the difference of squares \( (a-b)(a+b) = a^2 - b^2 \), the middle terms cancel out, leaving us with only the squared terms. This simplification is a key advantage when working with algebraic expressions because it allows us to compute products quickly and with less room for error.

Simplifying Equations

Simplifying equations is about reducing complexity to make an equation easier to understand and solve. This process includes combining like terms, factoring, canceling terms when possible, and applying algebraic rules. In the context of our problem, using the difference of squares formula simplifies the product of two binomials into a single expression without the need for long multiplication.

To illustrate, \( (7xy^2)^2 - (10y)^2 \) becomes \( 49x^2y^4 - 100y^2 \). This final form is much simpler than if we expanded the binomials traditionally using the FOIL method. Simplification is a powerful tool in algebra that, when mastered, can significantly speed up problem-solving and provide clearer insight into the structure of equations.