Problem 24 - Algebra 1: Concepts and Skills

Question

Write the product of the sum and difference. $$ (y-1)(y+1) $$

Step-by-Step Solution

Answer

\The short answer is \(y^2 - 1\).

Step 1: Identify the Numbers

The two numbers that are both being added and subtracted are `y` and `1`.

Step 2: Apply 'Difference of Squares' Identity

Substitute `y` and `1` into the formula \(a^2 - b^2\) to get \(y^2 - 1^2\).

Step 3: Simplify The Result

The expression simplifies to \(y^2 - 1\) because \(1^2 = 1\).

Key Concepts

Algebraic Identities

Algebraic identities are powerful tools in mathematics that help simplify expressions quickly by recognizing patterns. One of the most common algebraic identities is the "difference of squares" identity. This identity states that for any two numbers, \(a\) and \(b\), the expression \((a-b)(a+b)\) can be simplified to \(a^2 - b^2\).
This identity comes in handy when dealing with polynomial expressions, as it allows for a straightforward simplification process without requiring lengthy multiplication. Practically, it recognizes that multiplying a sum and a difference results in the square of the first term minus the square of the second term.
Understanding and identifying such patterns enable quicker problem-solving and a deeper conceptual grasp of algebraic operations.

Factoring Expressions

Factoring expressions is a key skill in algebra that involves breaking down polynomials into simpler components that multiply together to form the original expression. This is incredibly useful for solving equations and simplifying complex expressions.
The "difference of squares" serves as an excellent example of how factoring can streamline solving processes. In this case, the expression \(y^2 - 1\) can be seen as a difference of two squares: \(y^2\) is the square of \(y\), and \(1\) is the square of \(1\). As a result, the given expression \((y-1)(y+1)\) is naturally factored form of \(y^2-1\).
Being adept at spotting how polynomials can be rewritten using these identities aids in resolving problems more easily and readily prepares you for higher-level algebraic concepts.

Polynomial Simplification

Polynomial simplification is a process where an expression is reduced to its simplest form. This could involve combining like terms, applying algebraic identities, or performing arithmetic operations to render an expression more manageable.
In our example, polynomial simplification is achieved by applying the difference of squares identity. We started with \((y-1)(y+1)\), and by recognizing the identity, simplified it to \(y^2-1\).
Ultimately, simplifying polynomials makes them easier to work with, especially when solving equations, graphing functions, or performing further algebraic manipulation. Thus, mastering simplification techniques is crucial in efficiently tackling a wide array of algebraic challenges.