Problem 84 - Algebra and Trigonometry

Question

In Exercises 83–90, perform the indicated operation or operations.. $$ (5 x+2 y)^{2}-(5 x-2 y)^{2} $$

Step-by-Step Solution

Answer

The simplified equation is \(40xy\)

Step 1: Apply the difference of squares formula

Start by applying difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\). Here, \(a = 5x + 2y\) and \(b = 5x - 2y\). Plug a and b into the formula.

Step 2: Simplify the result

After applying the formula, simplify the result. The result becomes \((5x + 2y + 5x - 2y)(5x + 2y - 5x + 2y)\)

Step 3: Simplify again

Simplify the expressions in the brackets further to get the final result. It simplifies to \(10x * 4y\).

Step 4: Multiply the simplified equation

Multiply the numbers in the equation to get the final result which is \(40xy\)

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). Algebraic expressions can vary from simple forms like the numeric term 3x, to more complex formulations involving both numbers and variables raised to different powers.

When dealing with exercises like (5x + 2y)^2 - (5x - 2y)^2, you are essentially operating with algebraic expressions. In this particular case, both parts of the expression before subtraction represent perfect squares, which is a fundamental concept often utilized in simplifying problems involving algebraic expressions.

To ensure students fully grasp the subject, it's important to identify and understand the structure of these expressions. Recognizing patterns, such as the difference of squares, can simplify the process of manipulating and simplifying these expressions substantially.

Factoring Polynomials

To factor a polynomial means to break it down into simpler terms (the factors) that when multiplied together will give you the original polynomial. One of the key methods used in factoring is recognizing special products, such as a difference of squares.

For our exercise, the expression (5x + 2y)^2 - (5x - 2y)^2 falls under this special product category. By identifying this pattern, a seemingly complex problem is reduced to a more manageable one. The difference of squares is given by the formula \(a^2 - b^2 = (a + b)(a - b)\). When an expression mirrors this form, it can be factored quickly by using this formula.

Factoring is not just a mathematical sleight of hand; it allows students to simplify algebraic expressions that lead to easier calculation or further analysis, such as solving equations or graphing functions.

Simplifying Expressions

The term simplifying expressions refers to the process of reducing a mathematical expression into its simplest form. This often includes combining like terms, reducing fractions, and factoring out common factors among terms. By doing this, expressions become less complicated and calculations or evaluations can be performed with greater ease.

In the difference of squares example given above, once the original expression has been factored, the next critical step is to simplify the expression. This entails adding or subtracting the like terms and multiplying the constants, as we see in the given exercise where the middle terms cancel out and we are left with the simple product \(10x * 4y\), or \(40xy\).

Simplifying expressions is a cornerstone of algebra that ensures that calculations are efficient and solutions are presented clearly. Thus, helping students to accurately simplify their algebraic expressions is crucial for their ability to tackle more advanced mathematical problems effectively.