Problem 54 - Algebra 2

Question

Factor each expression completely. $$ 64 t^{2}-16 $$

Step-by-Step Solution

Answer

\((8t + 4)(8t - 4)\)

Step 1: Recognize the Difference of Squares

Notice that the expression can be written as the difference of two perfect squares: \(64t^2 = (8t)^2\) and \(16 = 4^2\). The expression \(64t^2 - 16\) is therefore written as \((8t)^2 - 4^2\).

Step 2: Apply the Difference of Squares Formula

Use the difference of squares formula, \(a^2 - b^2 = (a + b)(a - b)\), to factor the expression. Let \(a = 8t\) and \(b = 4\), and apply the formula to get \((8t + 4)(8t - 4)\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra, representing a combination of numbers, variables, and operations. These expressions are symbols that stand in for unknown values and can take on different forms, from simple constants to more complex polynomial expressions. An outstanding example includes expressions like the one in our textbook exercise, \(64t^2 - 16\), which combines constants and a variable raised to a power.

In working with algebraic expressions, one key skill is the ability to manipulate and rewrite them in various forms to simplify calculations or solve equations. This particular exercise demonstrates how an algebraic expression can be factored, which is a method of breaking down more complex expressions into simpler, multiplicative components.

Perfect Squares

A perfect square is the product of a number multiplied by itself. For instance, \(4^2 = 4 \times 4 = 16\) is a perfect square. In the context of our exercise, identifying that \(64t^2 - 16\) is a subtraction between two perfect squares \(\left(8t\right)^2\) and \(4^2\) is pivotal.

Recognizing perfect squares in algebraic expressions is crucial when it comes to simplifying or factoring them because it often unlocks the ability to apply specific factoring formulas. Awareness of common perfect squares, like \(1, 4, 9, 16, 25\), and so on, is incredibly useful during this process.

Factoring Polynomials

Factoring polynomials involves finding an equivalent expression that is a product of simpler polynomials. In simpler terms, it's like taking a more complex expression and 'breaking it apart' into factors that, when multiplied together, give you back the original expression.

For example, in our exercise, \(64t^2 - 16\) is factored into \(\left(8t + 4\right)\left(8t - 4\right)\). This process is akin to finding what numbers can be multiplied together to reach a given number but on a more abstract level— with variables and coefficients.

Difference of Squares Formula

The difference of squares formula is a key tool in factoring. It expresses that the difference between two perfect squares \(a^2 - b^2\) can be factored into \(\left(a + b\right)\left(a - b\right)\). This formula is particularly useful, as demonstrated in the exercise where \(64t^2 - 16\) becomes \(\left(8t\right)^2 - \left(4\right)^2\) and then is factored into \(\left(8t + 4\right)\left(8t - 4\right)\).

Understanding and applying the difference of squares formula requires identifying when an expression fits the pattern, a crucial step in quickly and efficiently factoring polynomials in algebra. By mastering this formula, students can solve a wide range of problems with confidence and ease.