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Chapter 6: Problem 85

Factor completely. $$ 8 x^{3}+125 \quad[5.6] $$

Short Answer

Expert verified
\(8x^3 + 125 = (2x + 5)(4x^2 - 10x + 25)\).

Step by step solution

01

Identify the form

Recognize that the expression is in the form of a sum of cubes: \(a^3 + b^3\). With this recognition, we can apply the sum of cubes factoring formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
02

Rewrite as a sum of cubes

Write each term as a cube: \[8x^3 = (2x)^3 \text{ and } 125 = 5^3\]. So the expression takes the form \((2x)^3 + 5^3\).
03

Apply the sum of cubes formula

Using the formula \((a^3 + b^3 = (a + b)(a^2 - ab + b^2)),\) where \(a = 2x\) and \(b = 5\), substitute these values into the equation: \[ (2x + 5)((2x)^2 - (2x)(5) + 5^2) \]
04

Simplify the factors

Now, simplify the expression inside the parentheses: \[ (2x + 5)(4x^2 - 10x + 25) \].

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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 algebraic form that appears frequently in polynomial expressions. It follows the formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
In the given exercise, we identify the expression \(8x^3 + 125\) as a sum of cubes, because both terms can be written as cubes: \(8x^3 = (2x)^3\) and \(125 = 5^3\).
Recognizing the sum of cubes form helps in applying the correct factoring formula to simplify the polynomial.
factoring formulas
Factoring formulas are crucial in simplifying polynomial expressions. They transform complex expressions into more manageable ones.
The sum of cubes formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) is just one example.
These formulas work by breaking down a polynomial into factors that are easier to handle and solve.
Using these tools correctly allows us to simplify expressions, like in the exercise, where we factor \(8x^3 + 125\) into \( (2x + 5)(4x^2 - 10x + 25)\).
polynomial expressions
Polynomial expressions are algebraic expressions consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents.
In the exercise, \(8x^3 + 125\) is a polynomial expression composed of two terms.
Understanding the basic structure of polynomial expressions is essential for applying specific factoring techniques.
algebra techniques
Algebra techniques are methods used to manipulate and simplify algebraic expressions.
Key techniques include recognizing patterns, applying factoring formulas, and simplifying expressions.
In the exercise, we used the technique of identifying the sum of cubes and then applied the corresponding formula.
This involves rewriting the polynomial as a sum of cubes and then simplifying the factors: from \(8x^3 + 125 = (2x)^3 + 5^3\) to finally \((2x + 5)(4x^2 - 10x + 25)\).

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Most popular questions from this chapter

For each pair of functions fand \(g,\) find all values of a for which \(f(a)=g(a)\) $$ \begin{array}{l}{f(x)=\frac{2}{x^{2}-8 x+7}} \\ {g(x)=\frac{3}{x^{2}-2 x-3}-\frac{1}{x^{2}-1}}\end{array} $$

Explain the difference between adding rational expressions and solving rational equations.

Factor completely. $$ z^{12}-1 \quad[5.6] $$

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Solve. $$ \frac{\frac{1}{3}}{x}=\frac{1-\frac{1}{x}}{x} $$
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