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Chapter 6: Problem 17

Factor completely. $$ 27 x^{3}+1 $$

Short Answer

Expert verified
\((3x + 1)(9x^2 - 3x + 1)\)

Step by step solution

01

Recognize the Sum of Cubes

Recognize that the expression is in the form of a sum of cubes. The given expression is \(27x^3 + 1\), which can be written as \((3x)^3 + 1^3\).
02

Apply the Sum of Cubes Formula

The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). In this case, identify \(a = 3x\) and \(b = 1\).
03

Substitute and Simplify

Substitute the identified values into the formula. This gives: \[(3x)^3 + 1^3 = (3x + 1)((3x)^2 - (3x)(1) + 1^2)\]
04

Calculate Each Term

Calculate each term in the factored form: \( (3x)^2 = 9x^2\), \((3x)(1) = 3x\) and \(1^2 = 1\).
05

Write the Complete Factored Form

Substitute the calculated terms back into the expression to get the complete factored form:\[(3x + 1)(9x^2 - 3x + 1)\]

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Key Concepts

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

Sum of Cubes
The sum of cubes is a special form of factoring where an expression in the form of \(a^3 + b^3\) is broken down. In this exercise, the expression \(27x^3 + 1\) is recognized as a sum of cubes. We can rewrite it as \((3x)^3 + 1^3\). By identifying \a = 3x\ and \b = 1\, we align the expression with the sum of cubes formula. Understanding the sum of cubes is crucial for identifying the correct terms and simplifying the expression efficiently. The sum of cubes formula, \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\), simplifies the process.
Factoring Formulas
Factoring formulas help in transforming complex algebraic expressions into simpler, factored forms. One of the key formulas used in this exercise is the sum of cubes formula. The formula is: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). By identifying the components \a\ and \b\, we can substitute them to get a factored expression. For \27x^3 + 1\, we used \a = 3x\ and \b = 1\. Substituting these into the formula, we get: \((3x) + 1)((3x)^2 - (3x)(1) + 1^2)\). This simplification process applies the foundational understanding of factoring techniques.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. Factoring them often requires recognizing patterns and applying formulas. For example, \(27x^3 + 1\) is rewritten as \((3x)^3 + 1^3\) to fit the sum of cubes pattern. Operations like squaring and multiplying terms are pivotal. In our example, we calculated each term in the factored form: \( (3x)^2 = 9x^2\), \((3x)(1) = 3x\), and \(1^2 = 1\). These steps exemplify how detailed calculation and substitution are vital for simplifying algebraic expressions effectively.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ -80 a^{6}+45 $$

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Write an equivalent expression by factoring. Assume that all exponents are natural numbers. $$ 3 a^{n+1}+6 a^{n}-15 a^{n+2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 16-a^{2}-2 a b-b^{2} $$
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