Question

Sum or Difference of Two Cubes. $$x^{3}+1$$

Short answer

\(x^{3}+1 = (x+1)(x^{2}-x+1)\)

Step-by-step solution

Identify the Sum of Two Cubes

Recognize that the given expression can be written as a sum of two cubes: \(x^{3}+1^3\). The general formula for the sum of two cubes is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here \(a = x\) and \(b = 1\).

Apply the Sum of Two Cubes Formula

Using the formula for the sum of two cubes, rewrite the expression as follows: \(x^{3}+1^3 = (x+1)(x^2-x(1)+1^2)\).

Simplify the Factored Form

Simplify the factored form without the placeholder multiplication by 1, to get: \((x+1)(x^2-x+1)\).

Key concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex polynomial into simpler ‘factors’ that, when multiplied together, give you the original polynomial. It's akin to decomposing a number into its prime factors. However, with polynomials, we are looking for algebraic expressions that multiply to give the original polynomial.

One common scenario where polynomial factoring is used is with cubic functions, particularly when they're in the form of the sum or difference of two cubes. Recognizing patterns, such as a cubic term plus a constant, can indicate the sum of two cubes, such as in the example of the exercise, where we factorize the polynomial \(x^{3}+1\).

Understanding how to factor polynomials helps in simplifying expressions and solving equations. It's an essential skill for algebra students, as it simplifies problems and reveals underlying structure that can be exploited to solve algebraic puzzles.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain ordinary numbers, variables (like \(x\) or \(y\)), and operators (such as plus, minus, multiply, and divide). The purpose of algebraic expressions is to describe relationships and changes between quantities in a general form. They can be as simple as \(x+1\) or as complex as the sum of two cubes expression \(x^{3}+1^3\).

In the educational exercise given, the expression \(x^{3}+1\) is an algebraic expression that we are asked to factor. This is a refined skill because it requires the student to recognize a special structure within the expression, the sum of cubes, and then apply a specific formula to break it apart into simpler expressions. This allows for a deeper understanding of the expression's properties and provides a foundation for more advanced operations such as simplification or equation solving.

Cubic Functions

Cubic functions are polynomial functions of degree three; that is, the highest power of the variable is three. Their general form is \(ax^{3}+bx^{2}+cx+d\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a \eq 0\). These functions can take on various shapes but typically have a point of inflection where the function changes concavity.

Cubic equations often present a more complex challenge when it comes to factoring, as they do not factor as readily as quadratic equations. But certain cubic expressions, like the sum or difference of two cubes, have a recognizable pattern that facilitates factoring. The exercise demonstrates this with the sum of two cubes, which uses the unique factoring formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) that can only be applied under the specific condition that the cubic function breaks down into a sum or difference of two perfect cubes.

Always remember, the properties of cubic functions are essential for understanding more complex algebraic expressions and are a vital step in developing mathematical problem-solving skills.