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Chapter 0: Problem 27

$$ \text { Factor the polynomials in Exercises } \text { . } $$ $$ 8 x^{3}+27 $$

Short Answer

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

Step by step solution

01

Identify the type of polynomial

This polynomial is a sum of cubes because it can be expressed in the form: a^3 + b^3.
02

Rewrite the polynomial as a sum of cubes

Rewrite the given polynomial as 8x^3 + 27 = (2x)^3 + 3^3.
03

Apply the sum of cubes formula

Use the sum of cubes formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
04

Substitute a and b

Here, a is 2x and b is 3. Substitute these values into the formula:(2x + 3)((2x)^2 - (2x)(3) + 3^2).
05

Simplify the expression

Simplify the terms inside the parenthesis:(2x + 3)(4x^2 - 6x + 9).
06

Write the final factored form

Thus, the factored form of the polynomial is: (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
Factoring polynomials often involves recognizing special forms, such as the sum of cubes. A polynomial is identified as a sum of cubes if it can be written in the form \(a^3 + b^3\). For instance, in the given exercise, we have \(8x^3 + 27\).

This expression fits the sum of cubes form, where \(a = 2x\) and \(b = 3\). By rewriting the polynomial, we get:

\[ 8x^3 + 27 = (2x)^3 + 3^3 \]

Knowing how to recognize these forms is crucial in polynomial factorization, as it guides the usage of appropriate algebraic formulas for factoring.
polynomial factorization
Polynomial factorization involves breaking down a polynomial into simpler 'factor' polynomials whose product gives the original polynomial. In this case, we use the sum of cubes formula to achieve this:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

For the polynomial \(8x^3 + 27\), we already identified \(a = 2x\) and \(b = 3\).

By substituting \(a\) and \(b\) into the formula, we get:

\[ (2x + 3)((2x)^2 - (2x)(3) + 3^2) \]

Next, simplify the terms inside the parentheses:

\[ (2x + 3)(4x^2 - 6x + 9) \]

Thus, the factored form of the polynomial \(8x^3 + 27\) is:

\[ (2x + 3)(4x^2 - 6x + 9) \]
algebraic formulas
Algebraic formulas are essential tools in mathematics, especially in polynomial factorization. In this exercise, we use the sum of cubes formula:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

This formula helps factor any polynomial of the form \(a^3 + b^3\).

Let's break down the formula:
  • \(a + b\): A binomial representing the sum of the two cube roots.
  • \(a^2 - ab + b^2\): A trinomial combining the squares and product of \(a\) and \(b\).


For our example, \(a = 2x\) and \(b = 3\). Thus,

\[ 8x^3 + 27 = (2x + 3)((2x)^2 - (2x)(3) + 3^2) \]

This simplifies to:

\[ (2x + 3)(4x^2 - 6x + 9) \]

Understanding and applying these formulas correctly helps in simplifying complex polynomials into more manageable parts.

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Most popular questions from this chapter

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