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Chapter 6: Problem 57

Factor each binomial completely. \(t^{3}+343\)

Short Answer

Expert verified
The binomial \(t^3+343\) factors to \((t+7)(t^2-7t+49)\).

Step by step solution

01

Identify Type of Expression

Recognize that the expression \(t^{3} + 343\) is a sum of cubes. This is because \(343\) can be expressed as \(7^3\), hence the binomial is \(t^3 + 7^3\).
02

Use the Sum of Cubes Formula

The formula for factoring a sum of cubes \(a^3 + b^3\) is \((a + b)(a^2 - ab + b^2)\). Here, \(a = t\) and \(b = 7\). Substitute these values into the formula.
03

Apply the Formula

Substitute \(a = t\) and \(b = 7\) into the sum of cubes formula: \((t + 7)(t^2 - 7t + 49)\). This gives us the completely factored form of the binomial.

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Key Concepts

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

Sum of Cubes
In algebra, certain expressions can be identified as the "sum of cubes". This refers to a specific type of expression that includes two terms, each of which is a cube. A cube, in mathematical terms, is any number raised to the power of three.

For instance, in our expression \(t^3 + 343\), both terms are cubes: \(t^3\) is \(t\) raised to the power of three, and 343 is the cube of 7, because \(7^3 = 343\).

Recognizing a sum of cubes is important because it allows us to use a particular formula to factor the expression completely.
Binomial
A binomial is a type of algebraic expression that contains two distinct terms. It's important to recognize these expressions, as they often have specific factoring techniques that can be applied. The expression \(t^3 + 343\) is an example of a binomial since it consists of two terms: \(t^3\) and 343.

Generally, when factoring binomials, you look for patterns or identities that fit the expression. For instance, the expression \(t^3 + 343\) fits the criteria for a sum of cubes, which guides us towards using the specific sum of cubes factoring formula to simplify and factor it further.
Algebraic Expression
Algebraic expressions are built from constants, variables, and algebraic operations (such as addition, subtraction, multiplication, and division). The expression \(t^3 + 343\) is a simple yet powerful example of an algebraic expression.

Understanding the structure of algebraic expressions, such as how to identify terms and degrees, is fundamental. In \(t^3 + 343\), the term \(t^3\) involves a variable \(t\) raised to the third degree, while 343 is a constant term, also expressed as \(7^3\).

Recognizing these components helps in applying the correct factoring methods.
Factoring Formulas
Factoring formulas are tools that help simplify algebraic expressions by breaking them down into products of simpler expressions. The sum of cubes formula is a specific factoring formula where \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).

This formula is invaluable for expressions like \(t^3 + 343\), allowing us to factor it into \((t + 7)(t^2 - 7t + 49)\).

Here's a quick breakdown of the operation:
  • Identify \(a\) and \(b\), where \(a = t\) and \(b = 7\).
  • Use the sum of cubes formula by substituting these values.
  • Complete the factoring to gain the simplest form of the expression.
Factoring formulas are essential, as they save time and effort while offering accurate solutions.

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Most popular questions from this chapter

Solve each equation. See Examples I and 2. $$ (2 x+3)(4 x-5)=0 $$

Factor each binomial completely. \(x^{2}-225 y^{2}\)

Multiply. See Section 5.4 $$ (5 y-4)(3 y-1) $$

Factor each trinomial completely. See Examples 1 through 11 and Section 6.2. \(-27 t+7 t^{2}-4\)

Find the GCF for each list. See Examples I through 3. $$ 14 x, 21 $$
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