Problem 36 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(5 y-2)(5 y+2)$$

Step-by-Step Solution

Answer

The product is \(25y^2 - 4\).

Step 1: Identify the Binomial Pattern

Recognize that the expression \((5y - 2)(5y + 2)\) fits the pattern of the difference of squares \((a-b)(a+b) = a^2 - b^2\). Here, \(a = 5y\) and \(b = 2\).

Step 2: Apply the Difference of Squares Formula

Substitute \(a = 5y\) and \(b = 2\) into the difference of squares formula, giving us \((5y)^2 - (2)^2\).

Step 3: Calculate Each Square

Calculate \((5y)^2\), which is \((5y)(5y) = 25y^2\), and \(2^2\), which is \(4\).

Step 4: Subtract the Squares

Subtract the smaller square from the larger one: \(25y^2 - 4\). This is the simplified form of the expression.

Key Concepts

Difference of Squares

The difference of squares is a special algebraic pattern that simplifies the multiplication of specific binomials. When you see an expression like \((a-b)(a+b)\), it can be transformed using the formula \(a^2 - b^2\). This pattern arises because multiplying opposites leads to the cancellation of the middle terms in the expansion.

• For example, with \((5y - 2)(5y + 2)\), the expression fits the difference of squares pattern.
• Here, \(a = 5y\) and \(b = 2\).
• Applying the formula, you get \((5y)^2 - (2)^2 = 25y^2 - 4\).

Understanding the difference of squares helps simplify expressions quickly without going through detailed multiplication of each term. It is especially useful in recognizing patterns and saving time in computations.

Simplifying Expressions

Simplifying expressions is about making them more manageable, usually by reducing them to fewer terms or simpler forms. When you simplify using algebraic patterns, you aim to express the equation as cleanly and efficiently as possible.

• You can see an example of this in the expression \(25y^2 - 4\), which is the result after applying the difference of squares.
• This means no further combination or factorization is necessary; you've reached a simplified and final state.
• Sometimes, simplifying may also involve techniques like combining like terms or reducing fractions to their simplest form.

Mastering this concept brings clarity to algebra and ensures you can handle more complex problems with ease.

Algebraic Patterns

Algebraic patterns are consistent, predictable arrangements in mathematical expressions that help us identify the most efficient processing method.

• The difference of squares is one such pattern, allowing for neat solutions to certain binomial multiplications.
• Recognizing these patterns instantly can aid in streamlining mathematical operations.
• For students, practicing these allows them to notice other algebraic patterns like the perfect square trinomial or the sum/difference of cubes.

Identifying these patterns in expressions not only makes problem-solving easier but also enhances the overall understanding of algebraic principles. With practice, you'll become adept at seeing the bigger picture and manipulating expressions effectively.