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Chapter 0: Problem 27

Factor the polynomials. $$8 x^{3}+27$$

Short Answer

Expert verified
\((2x + 3)(4x^2 - 6x + 9)\)

Step by step solution

01

Identify the Form

Recognize that the polynomial can be written as a sum of cubes. The given polynomial is \(8x^3 + 27\), which can be expressed as \((2x)^3 + 3^3\).
02

Use the Sum of Cubes Formula

Recall the sum of cubes formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). For the polynomial, let \(a = 2x\) and \(b = 3\).
03

Apply the Values to the Formula

Substitute \(a = 2x\) and \(b = 3\) into the formula: \((2x + 3)((2x)^2 - (2x)(3) + 3^2)\).
04

Simplify the Expression

Simplify the expression inside the parentheses: \((2x + 3)(4x^2 - 6x + 9)\).
05

Final Factored Form

The polynomial \(8x^3 + 27\) is factored as \((2x + 3)(4x^2 - 6x + 9)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Cubes Formula
When you encounter a polynomial of the form \text{a}^3 + \text{b}^3, you can use the sum of cubes formula to factor it. The sum of cubes formula is: \text{a}^3 + \text{b}^3 = (a + b)(a^2 - ab + b^2). This formula allows you to break down the polynomial into simpler components. In the given exercise, we recognized that \(8x^3 + 27\) is a sum of cubes because \(8x^3 = (2x)^3\) and \(27 = 3^3\). By setting \(a = 2x\) and \(b = 3\), we applied the formula to get: \((2x + 3)((2x)^2 - (2x)(3) + 3^2)\). It's always helpful to memorize this formula for quickly solving problems involving the sum of cubes.
Identifying Polynomial Forms
To factor polynomials correctly, it's essential to identify their forms. Different forms have different factoring rules. Common polynomial forms include:
  • Sum of cubes: \(a^3 + b^3\)
  • Difference of cubes: \(a^3 - b^3\)
  • Quadratic form: \(ax^2 + bx + c\)
Identifying the form of a polynomial is the first step in factoring it. In our exercise, recognizing \(8x^3 + 27\) as a sum of cubes was crucial. We saw \(8x^3\) and \(27\) and understood they could be rewritten as \((2x)^3\) and \(3^3\). This identification allows you to apply the right formula and simplify the problem quickly.
Algebraic Simplification
Simplifying algebraic expressions involves reducing them to their simplest form. After applying a formula, you often need to perform further simplifications. For the polynomial \(8x^3 + 27\), applying the sum of cubes formula gave us \((2x + 3)((2x)^2 - (2x)(3) + 3^2)\). To simplify:
  • Calculate \((2x)^2\) to get \(4x^2\)
  • Compute \((2x)(3)\) to get \(6x\)
  • Square \(3\) to get \(9\)
Putting it all together, \((2x + 3)(4x^2 - 6x + 9)\) is the simplified form. Always check each step to prevent errors. Simplification ensures the expression is in its most understandable and usable form.

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Most popular questions from this chapter

Let \(f(x)=\frac{x}{x-2}, g(x)=\frac{5-x}{5+x},\) and \(h(x)=\frac{x+1}{3 x-1} .\) Express the following as rational functions. $$h\left(\frac{1}{x^{2}}\right)$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$x^{5} \cdot\left(\frac{y^{2}}{x}\right)^{3}$$

Use your graphing calculator to find the value of the given function at the indicated values of \(x .\) $$f(x)=\frac{1}{2} x^{2}+\sqrt{3} x-\pi ; \quad x=-2, x=20$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{x^{3}}{y^{-2}}$$

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Calculate the following functions. Take \(x > 0\). $$\frac{f(x)}{g(x)}$$
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