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Chapter 10: Problem 51

Factor completely. $$ 8 t^{4}-8 t $$

Short Answer

Expert verified
The fully factored form is \( 8t(t - 1)(t^2 + t + 1) \).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, look for the greatest common factor (GCF) of the terms in the polynomial. Here, both terms, \(8t^{4}\) and \(8t\), have a common factor of \(8t\).
02

Factor Out the GCF

Next, factor out the GCF from both terms. This gives: \[ 8t(t^3 - 1) \]
03

Recognize the Difference of Cubes

The expression inside the parentheses is \(t^3 - 1\), which is a difference of cubes. Recall the difference of cubes formula: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
04

Apply the Difference of Cubes Formula

Rewrite \(t^3 - 1\) as \(t^3 - 1^3\), and apply the difference of cubes formula: \[ t^3 - 1 = (t - 1)(t^2 + t \cdot 1 + 1^2) = (t - 1)(t^2 + t + 1) \]
05

Complete the Factoring

Combine everything to get the fully factored form of the expression: \[ 8t(t^3 - 1) = 8t(t - 1)(t^2 + t + 1) \]

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Key Concepts

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

Greatest Common Factor
To start factoring any polynomial, you should first look for the greatest common factor (GCF). The GCF is the largest factor that divides all terms of the polynomial without leaving a remainder.
For example, in our polynomial:
  • 8t4 has factors 8, t, t2, t3, and t4.
  • 8t has factors 8 and t.
We notice that both terms share a common factor of 8t. So, the GCF is 8t. Extracting the GCF simplifies the polynomial and makes it easier to work with:
  • Given: 8t4 - 8t,
  • Factored form: 8t(t3 - 1).
This step is crucial as it lays the groundwork for further factoring.
Difference of Cubes
Now, let's tackle the expression inside the parentheses: t3 - 1. Recognize that this is a 'difference of cubes'.
The difference of cubes formula is: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Here, a = t and b = 1, so:
  • Rewrite t3 - 1 as t3 - 13
  • Using the formula: t3 - 1 = (t - 1)(t2 + t \times 1 + 12)
  • Simplified further: (t - 1)(t2 + t + 1).
Applying this formula can significantly simplify seemingly complex polynomials into more manageable factors.
Polynomials
Understanding polynomials is key to mastering factoring. A polynomial is an algebraic expression consisting of variables and coefficients, connected through addition, subtraction, multiplication, and positive integer exponents.
For instance, in the given exercise, 8t4 - 8t is a polynomial with two terms.
This polynomial is in standard form, arranged in decreasing powers of t.
Steps to factor:
  • Identify the polynomial's terms: 8t4 and -8t.
  • Find and extract the GCF: 8t(t3 - 1).
  • Recognize and apply special formulas like the difference of cubes when applicable.
By breaking it into smaller parts, understanding and solving polynomial problems becomes far less daunting!

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