Problem 47
Question
Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$x^{3}+1$$
Step-by-Step Solution
Verified Answer
\((x + 1)(x^2 - x + 1)\)
1Step 1: Identify the Pattern
The expression is given as \(x^3 + 1\). This is a sum of two cubes because it can be rewritten as \(x^3 + 1^3\). Recognize that it follows the sum-of-two-cubes pattern.
2Step 2: Recall the Formula for the Sum of Two Cubes
The sum of two cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 1\).
3Step 3: Substitute into the Formula
Substitute \(a = x\) and \(b = 1\) into the formula: \((x + 1)(x^2 - x \times 1 + 1^2)\).
4Step 4: Simplify the Expression
Simplify the expression: \((x + 1)(x^2 - x + 1)\). There are no further term simplifications possible, so this is the completely factored form.
Key Concepts
Sum of CubesAlgebraic ExpressionsPolynomialsMathematical Patterns
Sum of Cubes
The sum of cubes is an important concept in algebra that helps in factoring certain types of polynomials. When you have an expression of the form \(a^3 + b^3\), you're dealing with a sum of cubes. Identifying this allows you to use a specific pattern to factor the expression neatly and easily. The formula for the sum of two cubes is as follows:
\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
In our exercise, \(x^3 + 1\), we can see it as \(x^3 + 1^3\). Here, \(a = x\) and \(b = 1\). Plug these into the formula to break down the expression into simpler factors, making it much easier to handle. You are turning a complex-looking polynomial into a product of two simpler expressions.
\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
In our exercise, \(x^3 + 1\), we can see it as \(x^3 + 1^3\). Here, \(a = x\) and \(b = 1\). Plug these into the formula to break down the expression into simpler factors, making it much easier to handle. You are turning a complex-looking polynomial into a product of two simpler expressions.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and the operations of addition, subtraction, multiplication, or division. They are a fundamental part of algebra and mathematics in general, allowing mathematicians and students to represent real-world problems in a formal language. These expressions come in different types and can be combined, simplified, or factored.
When dealing with algebraic expressions like \(x^3 + 1\), recognizing patterns like the sum of cubes saves time and effort. By identifying the structure and form of the expression, you can apply relevant mathematical theories and patterns to factor the expression efficiently.
When dealing with algebraic expressions like \(x^3 + 1\), recognizing patterns like the sum of cubes saves time and effort. By identifying the structure and form of the expression, you can apply relevant mathematical theories and patterns to factor the expression efficiently.
Polynomials
Polynomials are special types of algebraic expressions that consist of variables raised to whole-number exponents and their coefficients. They can have multiple terms, and each term consists of a coefficient and a variable raised to an exponent. The term "polynomial" covers a broad range of expressions, from simple ones like \(x + 1\) to complex ones such as \(x^3 + 3x^2 + 3x + 1\).
In the context of factoring, recognizing a polynomial as one that can be expressed as a sum or difference of cubes enables you to apply specific factoring techniques. Polynomials like \(x^3 + 1\) are particularly straightforward to factor once you've identified their cube nature. This reduces them to products of binomials and trinomials, making them simpler to manipulate.
In the context of factoring, recognizing a polynomial as one that can be expressed as a sum or difference of cubes enables you to apply specific factoring techniques. Polynomials like \(x^3 + 1\) are particularly straightforward to factor once you've identified their cube nature. This reduces them to products of binomials and trinomials, making them simpler to manipulate.
Mathematical Patterns
Mathematical patterns, like the sum of cubes, play a significant role in simplifying and solving expressions in algebra. They provide strategies to identify and exploit structures within algebraic expressions. Patterns allow mathematicians and students to transform complex problems into simpler forms, making them easier to understand and calculate.
Using these patterns requires recognizing similarities and differences among polynomials. For example, the sum-of-cubes pattern helped us factor \(x^3 + 1\) into simpler parts by capitalizing on a well-known mathematical rule. Learning these patterns enhances problem-solving skills and helps in mastering algebraic manipulation.
Using these patterns requires recognizing similarities and differences among polynomials. For example, the sum-of-cubes pattern helped us factor \(x^3 + 1\) into simpler parts by capitalizing on a well-known mathematical rule. Learning these patterns enhances problem-solving skills and helps in mastering algebraic manipulation.
Other exercises in this chapter
Problem 47
Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$2 x^{2}+x-3=0$$
View solution Problem 47
Factor completely each of the polynomials and indicate any that are not factorable using integers. $$t^{4}-5 t^{2}+6$$
View solution Problem 47
Factor by grouping. $$a x+4 x+a y+4 y$$
View solution Problem 47
Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(6 x+7)(3 x-10)$$
View solution