Problem 23 - Elementary and Intermediate Algebra

Question

Factor. $$ 8+t^{3} $$

Step-by-Step Solution

Answer

Factor using the sum of cubes: \((2 + t)(4 - 2t + t^2)\).

Step 1: Identify the Expression to Factor

We need to factor the expression: \(8+t^{3}\).

Step 2: Look for Common Factors and Patterns

We examine the expression to find common factors, patterns (like difference of squares, perfect square trinomials, grouping), or use appropriate factoring techniques.

Step 3: Write the Factored Form

Factor using the sum of cubes: \((2 + t)(4 - 2t + t^2)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that involves breaking down a polynomial into simpler components, or factors, which, when multiplied together, give the original polynomial. In simpler terms, it's like breaking down a big number into the smaller numbers that multiply together to form it. To factor expressions like polynomials, you first need to identify if there are any common factors across the terms. Then, depending on the structure and number of terms, different methods can be applied:

• Common factoring
• Factoring by grouping
• Using special formulas such as the sum and difference of squares, cubes, etc.

For instance, in our original exercise, the expression was identified as a sum of cubes, which requires applying a specific formula dedicated to this type of factoring. Recognizing the form of the polynomial is crucial because it dictates the strategy to use. Understanding this concept not only helps in simplifying expressions but also in solving equations, as factoring allows us to set each factor equal to zero in order to find solutions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations. They allow us to summarize complex mathematical ideas. Unlike equations, they do not have an equal sign.For instance, in the given exercise, "\(8 + t^3\)" is an algebraic expression. It includes constants (like 8), variables (like \(t\)), and operations (addition and exponentiation). Understanding algebraic expressions requires familiarity with terms:

• Terms: The individual components of an expression (e.g., \(8\) and \(t^3\)).
• Coefficients: Numbers multiplying the variables (though not explicitly in "\(t^3\)", we can assume a coefficient of 1).
• Operations: The mathematical operations applied between terms (like addition, subtraction, multiplication).

Mastering how to work with these expressions is foundational to understanding more complex algebraic processes, such as factoring and solving polynomial equations.

Sum of Cubes Formula

The sum of cubes formula is a special algebraic identity used when two cubes are added together in an expression. The generic form of the sum of cubes is \(a^3 + b^3\). To factor this, we use the formula:\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]In the context of the exercise, the expression "\(8 + t^3\)" can be seen as "\(2^3 + t^3\)". Here, \(a = 2\) and \(b = t\). Applying the sum of cubes formula gives us:\[(2 + t)(2^2 - 2 \cdot t + t^2)\]Breaking it down further:

• The first term, \(2 + t\), is straightforward.
• The second term, \(4 - 2t + t^2\), involves simple multiplication and calculation of squares.

This approach helps in simplifying and rewriting expressions into a product of factors, which is often easier to work with, especially in solving equations. Understanding how to identify and apply this formula can save time and provide insight into the structure of the expression itself.