Problem 85 - Beginning and Intermediate Algebra

Question

Find the following special products. $$(2-r)(2+r)$$

Step-by-Step Solution

Answer

The special product of \((2-r)(2+r)\) is \(4 - r^2\).

Step 1: Identify A and B

First, let's identify A and B, which represent two expressions in the formula: \[A^2 - B^2 = (A-B)(A+B)\] In our given expression \((2-r)(2+r)\), we can identify as follows: - A = 2 - B = r

Step 2: Apply the Difference of Squares Formula

Now that we have identified A and B, we can directly substitute the values in the formula: \[A^2 - B^2 = (A-B)(A+B)\] Substitute A = 2 and B = r: \[(2^2 - r^2) = (2-r)(2+r)\]

Step 3: Calculate the Square of A and B

Square the values of A and B: - A^2 = 2^2 = 4 - B^2 = r^2

Step 4: Calculate the Special Product

Now substitute A^2 = 4 and B^2 = r^2 back into the formula: \(4 - r^2 = (2-r)(2+r)\) Therefore, the special product of \((2-r)(2+r)\) is \(4 - r^2\).

Key Concepts

Special Products

Special products in algebra are expressions that simplify into particular forms. They're used because they appear frequently and recognizably, which makes them easy to handle. One common type of special product is the "difference of squares" which occurs when you have a product of the form

• \[(A-B)(A+B) = A^2 - B^2\]

This form is special because multiplying these binomials doesn't have multiple individual terms; it consolidates into just two terms. This simplicity is achieved due to the middle terms (the sum and the difference) canceling each other out. This specific trait makes problems involving special products quicker to solve, easing up complex expressions.

Factoring

Factoring in algebra involves breaking down expressions into simpler multiplicative components. When dealing with special products like the difference of squares, factoring becomes even more straightforward. For the expression

• \((2-r)(2+r)\),

we can think about reversing the simplification process from the special product form. Recognizing the pattern, we factor numbers or expressions back into their original binomial pairings, uncovering valuable insights about the expression. Factoring is a crucial skill because it simplifies complicated mathematical tasks and aids in solving equations.

Algebra

Algebra provides the language and structure that allows us to manipulate mathematical expressions. It's the branch of mathematics dealing with symbols and the rules for manipulating those symbols. The use of formulas like

• \(A^2 - B^2 = (A-B)(A+B)\)

illustrates how algebra unifies many problems under the same technique, offering consistency and logic. Algebra bridges arithmetic to more abstract mathematics, opening students to a world where variables and constants dance together in harmony, simplified and understood through established rules and formulas.

Mathematical Expressions

In mathematics, expressions are combinations of numbers, variables, and operation symbols, like addition and subtraction. Expressions don't include equality signs like equations do; they represent a value rather than a statement to solve.

• Consider \((2-r)(2+r)\).

This expression showcases the structure of the difference of squares. Simplifying such expressions involves identifying patterns or special products that help reduce it to a more straightforward form. Mastery over expressions ensures that students can handle algebraic manipulation efficiently, preparing them for complex problem-solving in advanced mathematics.