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Chapter 5: Problem 90

Factor each binomial completely. \(216 x^{3}+125\)

Short Answer

Expert verified
(6x + 5)(36x^2 - 30x + 25)

Step by step solution

01

Identify the Sum of Cubes

Recognize that the given expression is a sum of cubes. It can be written as: 216x^3 + 125 = (6x)^3 + 5^3.
02

Apply the Sum of Cubes Formula

For any binomials of the form a^3 + b^3, the sum of cubes can be factored using the formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
03

Substitute Values Into the Formula

In this case, let a = 6x and b = 5. Then plug these values into the sum of cubes formula: (6x)^3 + 5^3 = (6x + 5)((6x)^2 - (6x)(5) + 5^2).
04

Simplify the Expression

Now simplify the expression: (6x + 5)(36x^2 - 30x + 25). Thus: 216x^3 + 125 can be factored completely as (6x + 5)(36x^2 - 30x + 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' refers to an algebraic expression where two cubed terms are added together. In our example, we have 216x^3 + 125. To recognize this sum of cubes, we rewrite it as (6x)^3 + 5^3.
Whenever you see an expression in the form a^3 + b^3, remember that this is a sum of cubes. This sets us up to apply specific factoring formulas to simplify our work. Don't worry, it's simpler than it sounds because there is a direct formula to factor these sums.
Factoring formulas
Factoring formulas make simplifying expressions much easier. For the sum of cubes, there's a special formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
This formula helps break down seemingly complex expressions into simpler binomials and trinomials. For our problem, a is 6x and b is 5. Using the formula, we substitute 6x for 'a' and 5 for 'b'.
This gives us the factorization as: (6x + 5)((6x)^2 - (6x)(5) + 5^2).
This formula is very powerful and can save you a lot of time when solving similar problems.
Algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations. They are the foundation of algebra and appear everywhere in mathematics.
In our example, we start with the algebraic expression 216x^3 + 125. Recognizing patterns like the sum of cubes helps in making the problem manageable. Factoring this expression simplifies it to (6x + 5)(36x^2 - 30x + 25).
Understanding how to manipulate and simplify algebraic expressions through factoring is an essential skill. It will help you solve equations and understand deeper algebraic concepts. Practicing problems like these improve your ability to recognize patterns and apply the correct formulas.

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Most popular questions from this chapter

Factor the integer 35 as the product of the negatives of two prime numbers.

Solve each problem. Tram works due north of home. Her husband Alan works due east. They leave for work at the same time. By the time Tram is \(5 \mathrm{mi}\) from home, the distance between them is \(1 \mathrm{mi}\) more than Alan's distance from home. How far from home is Alan?

Which of the following are sums of cubes? A. \(x^{3}+1\) B. \(x^{3}+36\) C. \(12 x^{3}+27\) D. \(64 x^{3}+216 y^{3}\)

Factor each binomial completely. \(125 t^{3}+8 s^{3}\)

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