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Chapter 5: Problem 82

Factor each binomial completely. \(b^{3}+1\)

Short Answer

Expert verified
\(b^{3} + 1 = (b + 1)(b^{2} - b + 1)\)

Step by step solution

01

Recognize the Sum of Cubes

Identify the given expression as the sum of cubes. The expression is in the form of \(a^{3} + b^{3}\) where \(a = b\) and \(b = 1\).
02

Use the Sum of Cubes Formula

Apply the formula for the sum of cubes: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). Here, \(a = b\) and \(b = 1\).
03

Substitute Values

Substitute \(a = b\) and \(b = 1\) into the formula: \[b^{3} + 1 = (b + 1)(b^{2} - b \times 1 + 1^{2})\].
04

Simplify the Expression

Simplify inside the parentheses: \[b^{3} + 1 = (b + 1)(b^{2} - b + 1)\].

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Key Concepts

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

Sum of Cubes
The concept of 'Sum of Cubes' is central to solving binomials like the exercise given. In mathematics, a sum of cubes refers to an expression of the form \(a^{3} + b^{3}\). This type of expression can be factored using a specific formula, which is \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). The given exercise has \(b^{3} + 1\). Here, \(a\) is equivalent to \(b\) and \(b\) is equivalent to 1. Recognizing and identifying that a binomial fits the sum of cubes form is the first and most important step.
Algebraic Expressions
Understanding algebraic expressions is essential for working on factoring problems. An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. For instance, in the exercise provided, \(b^{3} + 1\) is an algebraic expression. It includes the variable \(b\) and the operation of addition. Converting these expressions into factored form involves using known formulas and substituting the values. This process simplifies the expression and helps with further algebraic manipulation or solving equations. Always remember the basic structure and how to manipulate these expressions when factoring.
Factoring Formulas
Factoring formulas are invaluable tools in algebra. They allow us to break down complex expressions into simpler, more manageable parts. In our exercise, we use the formula for the sum of cubes: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). Applying this formula requires identifying the 'a' and 'b' in the expression \(b^{3} + 1\), and then substituting them into the formula: \(a = b\) and \(b = 1\). This results in \( (b + 1)(b^{2} - b + 1)\). These formulas simplify the process and are critical for success in algebra.

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Most popular questions from this chapter

The following powers of \(x\) are all perfect cubes: $$ x^{3}, x^{6}, \quad x^{9}, \quad x^{12}, \quad x^{15} $$ On the basis of this observation, we may make a conjecture that if the power of a variable is divisible by ____ (with 0 remainder), then we have a perfect cube.

If an object is projected upward with an initial velocity of \(128 \mathrm{ft}\) per sec, its height \(h\) in feet after t seconds is given by the quadratic equation $$ h=-16 t^{2}+128 t $$ Find the height of the object after each time listed. \(2 \mathrm{sec}\)

Solve each problem. Find two consecutive odd integers such that their product is 15 more than three times their sum.

Find all integers \(k\) so that the trinomial can be factored using the methods of this section. \(2 x^{2}+k x-3\)

Apply the special factoring rules of this section to factor each polynomial. \(p^{2}-\frac{1}{9}\)
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