Problem 54
Question
Write the expression in factored form. \(27 x^{3}-8\).
Step-by-Step Solution
Verified Answer
The expression in factored form is \((3x - 2)(9x^2 + 6x + 4)\
1Step 1: Identify a and b
Given that the formula for difference of two cubes is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\), we first need to find \(a\) and \(b\) that satisfy \(a^3 = 27x^3\) and \(b^3 = 8\). Upon solving these two cubes, we find \(a = 3x\) and \(b = 2\)
2Step 2: Substitute a and b into the formula
We now substitute \(a = 3x\) and \(b = 2\) into the difference of cubes formula. Doing this, we get \((3x - 2)((3x)^2 + (3x)(2) + (2)^2)\)
3Step 3: Simplify the expression
The next step is to simplify the resulting expression. We get \((3x - 2)(9x^2 + 6x + 4)\) as the factored form of the expression
Key Concepts
Polynomial FactorizationAlgebraic IdentitiesSimplifying Expressions
Polynomial Factorization
Understanding the process of polynomial factorization is crucial when dealing with complex algebraic expressions. It involves breaking down a polynomial into a product of simpler polynomials or factors. This can simplify solving equations, graphing functions, and identifying roots. For instance, the expression \(27x^3 - 8\) can be factored using the difference of cubes, a special factorization method for polynomials that can be expressed as the difference of two cubic terms.
In polynomial factorization, it's important to recognize common patterns, such as the difference of squares and cubes. These patterns adhere to specific formulas that make factorization straightforward once the appropriate formula is identified. In our case, the difference of cubes is factored using the identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), where \(a\) and \(b\) are the cube roots of the respective terms in the original expression.
In polynomial factorization, it's important to recognize common patterns, such as the difference of squares and cubes. These patterns adhere to specific formulas that make factorization straightforward once the appropriate formula is identified. In our case, the difference of cubes is factored using the identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), where \(a\) and \(b\) are the cube roots of the respective terms in the original expression.
Algebraic Identities
Facilitating the simplification of algebraic expressions, algebraic identities are equations that are true for all values of the variables involved. They are powerful tools in algebra, serving as a shortcut to expand, factorize, or simplify expressions without having to perform more tedious or complex algebraic operations.
One such identity is the difference of cubes, integral to weaning down our example, \(27x^3 - 8\). This identity is especially useful as it applies to any cubic terms that can be structured as a difference, leading to \(a^3 - b^3\). Both educators and students should become familiar with these identities as they are recurring themes in algebra problems:
One such identity is the difference of cubes, integral to weaning down our example, \(27x^3 - 8\). This identity is especially useful as it applies to any cubic terms that can be structured as a difference, leading to \(a^3 - b^3\). Both educators and students should become familiar with these identities as they are recurring themes in algebra problems:
- \(a^2 - b^2 = (a - b)(a + b)\) - Difference of squares
- \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) - Sum of cubes
- \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) - Difference of cubes
Simplifying Expressions
The act of simplifying expressions is central to algebra and involves reducing an expression to its simplest form while maintaining its original value. This can be achieved through factorization, combining like terms, and applying algebraic identities among other techniques. Simplification makes the expressions easier to work with and often reveals deeper insights into their properties and solutions.
In our original problem \(27x^3 - 8\), simplification through the process of factoring difference of cubes leads to the expression \(3x - 2)(9x^2 + 6x + 4)\) where each factor is simpler than the original cubic terms. This not only makes the subsequent operations like finding roots or derivatives more manageable but also clarifies the underlying structure of the polynomial.
The ability to simplify complex expressions efficiently is a cornerstone of algebraic proficiency and can significantly benefit students across various mathematical tasks, from solving equations to analyzing functions.
In our original problem \(27x^3 - 8\), simplification through the process of factoring difference of cubes leads to the expression \(3x - 2)(9x^2 + 6x + 4)\) where each factor is simpler than the original cubic terms. This not only makes the subsequent operations like finding roots or derivatives more manageable but also clarifies the underlying structure of the polynomial.
The ability to simplify complex expressions efficiently is a cornerstone of algebraic proficiency and can significantly benefit students across various mathematical tasks, from solving equations to analyzing functions.
Other exercises in this chapter
Problem 54
Form the compositions \(f \circ g\) and \(g \circ f,\) and specify the domain of each of these combinations. $$f(x)=\sqrt{1-x^{2}}, \quad g(x)=\sin 2 x$$
View solution Problem 54
State whether the function is odd, even, or neither. $$F(x)=x+\frac{1}{x}$$
View solution Problem 54
Give the domain and range of the function. $$g(x)=\sin ^{2} x+\cos ^{2} x$$.
View solution Problem 54
Show that \(|a+b|=|a|+|b|\) iff \(a b \geq 0\).
View solution