Problem 57 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ t^{3}+1 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Expression Type

The expression you need to factor is \(t^3 + 1\). This is a sum of cubes, which follows the identity for factoring: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a\) is \(t\) and \(b\) is 1.

Step 2: Apply the Sum of Cubes Formula

Substitute \(a = t\) and \(b = 1\) into the sum of cubes formula. This gives us the factored form: \((t + 1)(t^2 - t \cdot 1 + 1^2)\). Simplifying, we get \((t + 1)(t^2 - t + 1)\).

Step 3: Verify the Factorization

To ensure the factorization is correct, expand \((t + 1)(t^2 - t + 1)\) back. First, expand \(t \cdot (t^2 - t + 1)\) to get \(t^3 - t^2 + t\). Then expand \(1 \cdot (t^2 - t + 1)\) to get \(t^2 - t + 1\). Adding these, \(t^3 - t^2 + t + t^2 - t + 1 = t^3 + 1\), confirms the factorization is accurate.

Key Concepts

Sum of Cubes

Understanding the sum of cubes is crucial when factoring expressions like \( t^3 + 1 \). The sum of cubes formula allows us to break down expressions of the form \( a^3 + b^3 \) into a product of two simpler algebraic expressions. The formula is given by:

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

In this formula, \( a \) and \( b \) represent the cube roots of the terms. When you encounter an expression like \( t^3 + 1 \), recognize \( a \) as \( t \) and \( b \) as \( 1 \). By substituting these values into the formula, you can factor the expression completely. This method provides an effective way to handle a variety of polynomial equations involving cubes.
It's also useful to remember that unlike some factoring techniques, the sum of cubes, unlike the difference of squares, doesn't result in two identical terms, but in a binomial and a trinomial.

Algebraic Expressions

Algebraic expressions are the building blocks of math, combining variables and constants in a meaningful way. An expression like \( t^3 + 1 \) involves variables raised to powers and terms that are combined with mathematical operations.

• Terms like \( t^3 \) are called polynomial terms where \( t \) is the variable raised to the power of 3.
• Constants, such as \(1\) in our expression, play a vital role in determining the structure of the expression.

Algebraic expressions can be simple linear terms or complex polynomials involving multiple variables and operations. The key to solving problems involving algebraic expressions is to understand how to manipulate them using mathematical operations including addition, subtraction, multiplication, and division.
This manipulation often includes factoring, which simplifies expressions to make calculations easier and to find solutions or roots of equations. Recognizing the type and structure of the expression is the first step in effective factorization.

Polynomial Expansion

Polynomial expansion is a technique used to simplify and verify the expressions obtained after factoring. When you factor an expression like \((t + 1)(t^2 - t + 1)\), expanding it helps confirm that it equals the original expression \(t^3 + 1\).

• Start by multiplying each term in the first bracket with every term in the second bracket.
• For example, multiply \(t\) by each term in \(t^2 - t + 1\) to obtain \(t^3 - t^2 + t\).
• Next, multiply \(1\) by each term in \(t^2 - t + 1\) to get \(t^2 - t + 1\).

Then, combine all these terms: \(t^3 - t^2 + t + t^2 - t + 1\). Simplifying this by adding like terms results in \(t^3 + 1\). This process not only verifies the factorization but also reinforces understanding of polynomial operations. Learning to expand polynomials accurately is a fundamental skill in algebra, as it supports various mathematical techniques, such as integration and differentiation in calculus.