Problem 61 - Algebra and Trigonometry

Question

Factor the expression completely. $$ t^{3}+1 $$

Step-by-Step Solution

Answer

The factored form is \((t + 1)(t^2 - t + 1)\).

Step 1: Identify the Formula for Sum of Cubes

The expression given, \(t^3 + 1\), is a sum of cubes since it can be written as \(t^3 + 1^3\). To factor a sum of cubes, we use the formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). In this case, \(a = t\) and \(b = 1\).

Step 2: Substitute the Values into the Formula

Substitute \(a\) and \(b\) into the sum of cubes formula. This gives us:\[(t^3 + 1) = (t + 1)(t^2 - t imes 1 + 1^2)\].

Step 3: Simplify the Expression

Now, we simplify the second factor:\[(t^2 - t imes 1 + 1^2) = (t^2 - t + 1)\].

Step 4: Write the Fully Factored Form

The fully factored form of \(t^3 + 1\) is \((t + 1)(t^2 - t + 1)\).

Key Concepts

Sum of Cubes

Polynomials often contain terms that are perfect cubes. A common scenario in algebra is the sum of cubes, which is represented as \( a^3 + b^3 \). In such cases, there is a specific formula that simplifies factoring these types of expressions. The formula is:

• \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

Using this formula, you can break down complex expressions into simpler parts. For instance, in the expression \( t^3 + 1 \), it can be rewritten as a sum of cubes because \( t^3 \) is a cube, and \( 1 = 1^3 \) is also a cube. This makes it straightforward to apply the sum of cubes formula with \( a = t \) and \( b = 1 \).
After substitution, you get:

• \((t + 1)(t^2 - t \times 1 + 1^2)\)

This takes the original sum of cubes and expresses it in a factorized form. Understanding and identifying when to use this formula is key to simplifying polynomial expressions.

Algebraic Expressions

An algebraic expression consists of numbers, operations, and variables that are combined together. In our example, \( t^3 + 1 \), we see an algebraic expression made up of a polynomial term \( t^3 \) and a constant \( 1 \). Expressions such as these can often be broken down further or manipulated for various purposes, such as simplification or solving equations.
Algebraic expressions are foundational to algebra because they can represent real-world problems in a structured, numerical form. They allow you to perform operations and simplify equations, helping you find unknown values or patterns. For instance, when you factorize \( t^3 + 1 \) using the sum of cubes, it breaks down the task of solving or simplifying complex algebraic problems into more manageable parts.

• They include:
• Variables like \( t \) in \( t^3 \)
• Constants like \( 1 \)
• Operations such as addition and multiplication

Knowing how to work with algebraic expressions is crucial, as it forms the basis of algebraic manipulation and problem solving.

Polynomial Factorization

Factorization is a powerful tool in algebra that involves breaking down complex expressions into simpler, multiplied components. In our example, \( t^3 + 1 \), this is a polynomial that can be factorized by using the formula for the sum of cubes. Factorization simplifies expressions, making them more manageable or easier to solve.
In polynomial factorization, you aim to express a polynomial as a product of its factors. For example, the polynomial \( t^3 + 1 \) becomes \( (t + 1)(t^2 - t + 1) \). This transformation not only simplifies the expression but sometimes reveals the roots or solutions of the equation.
The steps to factorizing a polynomial usually include identifying a pattern, applying an appropriate formula such as the sum or difference of cubes, and simplifying the expression into its factorized form. Understanding these patterns and applying the correct formulas is key to mastering polynomial factorization, reducing mathematical complexity, and solving equations effectively.