Problem 43 - Elementary and Intermediate Algebra

Question

Factor. If a polynomial can't be factored, write "prime." $$ t^{2}-25 $$

Step-by-Step Solution

Answer

\((t - 5)(t + 5)\)

Step 1: Identify the Type of Polynomial

Recognize that the given polynomial \( t^2 - 25 \) is a difference of squares. The general formula for factoring a difference of squares is \( a^2 - b^2 = (a-b)(a+b) \).

Step 2: Determine the Square Terms

Identify \( a^2 \) and \( b^2 \) in the polynomial \( t^2 - 25 \). Here, \( a^2 = t^2 \) and \( b^2 = 25 \). Notice that \( a = t \) and \( b = 5 \), since \( 25 \) is \( 5^2 \).

Step 3: Apply the Difference of Squares Formula

Use the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \) to factor the polynomial. Substituting \( a = t \) and \( b = 5 \), we get \( t^2 - 25 = (t - 5)(t + 5) \).

Step 4: Write the Factored Form

The factored form of the polynomial \( t^2 - 25 \) is \( (t - 5)(t + 5) \).

Key Concepts

Difference of Squares

One fundamental concept in mathematics is the difference of squares. It occurs when we have an expression such as \(a^2 - b^2\). This pattern is special because it can be easily factored into the expression \((a - b)(a + b)\). Factorization is simplifying algebraic expressions to their simplest components or form, and the difference of squares is a classic example of this.

• "Difference" means subtraction, so the expression involves two squares being subtracted from each other.
• The phrase "of squares" indicates that each of the terms involved is itself a square term.

Recognizing an expression as a difference of squares can greatly simplify the process of solving or simplifying algebraic expressions, as it allows direct application of the formula \( a^2 - b^2 = (a-b)(a+b) \). This technique is particularly useful when solving quadratic equations or simplifying complex algebraic formulas.

Polynomial Factoring

Polynomial factoring is an essential technique that simplifies expressions and solves equations. At its core, factoring means breaking down a polynomial into simpler polynomials whose product gives the original polynomial.
There are different methods and strategies for polynomial factoring, with the difference of squares being just one. For example, we might look for common factors or utilize specific patterns like perfect square trinomials or sum and difference of cubes.

• Recognizing patterns is crucial in deciding which factoring strategy to apply.
• Factorization reduces complex expressions to simpler terms, making them easier to handle in mathematical operations.

In the example \(t^2 - 25\), recognizing that it is a difference of squares allows us to factor it into two binomials: \((t-5)(t+5)\). Each factor gives insight into the solution or simplification of the expression. Understanding these strategies ensures that students do not overlook potential simplifications.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations that form meaningful calculations. They are the language of algebra and allow us to make generalizations and solve real-world problems. Often, these expressions can be simplified or transformed to reveal important properties or solutions.

• Expressions like \(t^2 - 25\) are composed of terms that include variables raised to powers, constants, or both.
• Operations such as addition, subtraction, multiplication, or division connect these terms.

Decomposing the original expression into simpler components by factoring, as in the example \((t-5)(t+5)\), is crucial. It not only simplifies but also unveils potential points of intersection, solutions, or key properties about the variables involved. This process of simplification through factoring is vital when dealing with quadratic expressions, whether for academic exercises or practical applications.