/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 A polynomial \(P\) is given. (... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 17

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{6}-1$$

Short Answer

Expert verified
Zeros are 1, -1, \(\frac{-1 \pm i\sqrt{3}}{2}\), \(\frac{1 \pm i\sqrt{3}}{2}\). Factor: \((x - 1)(x + 1)(x^2 + x + 1)(x^2 - x + 1)\).

Step by step solution

01

Recognize the Polynomial Form

The given polynomial is \(x^6 - 1\). This fits the form \(a^6 - b^6\), which can be factored using the difference of two squares and further factored by recognizing it as a difference of cubes and squares.
02

Apply the Difference of Squares

Start by applying the difference of squares: \(x^6 - 1 = (x^3)^2 - 1^2 = (x^3 - 1)(x^3 + 1)\). This will break the polynomial into two cubic factors which can be further factored.
03

Factor Each Cubic Expression

Factor each of the cubic terms:- \(x^3 - 1\) factors to \((x - 1)(x^2 + x + 1)\) using the formula for the difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).- \(x^3 + 1\) factors to \((x + 1)(x^2 - x + 1)\) using the formula for the sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
04

Form the Completely Factored Polynomial

Combine the factors from the cubic expressions to completely factor \(x^6 - 1\): \(P(x) = (x - 1)(x^2 + x + 1)(x + 1)(x^2 - x + 1)\).
05

Solve for the Zeros

Set each factor to zero to find all zeros of \(P(x)\).- \(x - 1 = 0\) gives \(x = 1\).- \(x + 1 = 0\) gives \(x = -1\).- Solve \(x^2 + x + 1 = 0\) using the quadratic formula: \[x = \frac{-1 \pm i\sqrt{3}}{2}\]- Solve \(x^2 - x + 1 = 0\) also using the quadratic formula: \[x = \frac{1 \pm i\sqrt{3}}{2}\]
06

Compile the Zeros and Factors

The real zeros are \(x = 1\) and \(x = -1\). The complex zeros stemming from \(x^2 + x + 1\) are: \(x = \frac{-1 \pm i\sqrt{3}}{2}\), and from \(x^2 - x + 1\) are: \(x = \frac{1 \pm i\sqrt{3}}{2}\). The factorized polynomial is \(P(x) = (x - 1)(x^2 + x + 1)(x + 1)(x^2 - x + 1)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Difference of Squares
The **difference of squares** is a powerful algebraic tool used to simplify expressions. It applies to any expression of the form \(a^2 - b^2\), which can be factored into \((a - b)(a + b)\). This simplification is used because it breaks down a degree-two polynomial into two linear terms.
For example, in the original problem, the expression \(x^6 - 1\) can be recognized as a difference of squares since it fits the form \((x^3)^2 - 1^2\).
This allows us to factor it into \((x^3 - 1)(x^3 + 1)\).

**Key Points:**
  • Recognize expressions in the form \(a^2 - b^2\).
  • Break them down into \((a - b)(a + b)\).
  • Factoring makes solving polynomials easier by reducing their degree.
Difference of Cubes
In algebra, the **difference of cubes** formula helps to factor expressions like \(a^3 - b^3\). The general formula is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
This method can simplify complex polynomials into more manageable parts. In the polynomial \(x^3 - 1\) from the problem, we apply the difference of cubes formula.
The term \(x^3 - 1\) can thus be factored as \((x - 1)(x^2 + x + 1)\).

**Key Points:**
  • Identify an expression that is a difference of cubes.
  • Use the formula, substituting \(a\) and \(b\) with appropriate values.
  • Factor into a linear term and a quadratic term.
Factoring these expressions not only simplifies finding zeros but also aids in the complete factorization of complex polynomials.
Complex Zeros
**Complex zeros** are the roots of a polynomial that are not real numbers, often involving imaginary units. When factoring polynomials, especially of higher degrees, it's common to encounter non-real roots.
The imaginary unit \(i\) satisfies \(i^2 = -1\), allowing the solution of quadratic equations that don't have real solutions.
For instance, when solving \(x^2 + x + 1 = 0\), the quadratic formula reveals complex zeros:
  • Apply the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • For \(x^2 + x + 1 = 0\), this results in \(x = \frac{-1 \pm i\sqrt{3}}{2}\).
Thus, \(x^2 + x + 1\) provides complex zeros, \(x = \frac{-1 \pm i\sqrt{3}}{2}\), that can factor the polynomial into expressions including these roots.
**Understanding Zeros:**
  • Real zeros are the roots located on the real number line.
  • Complex zeros include imaginary components and appear in conjugate pairs.
  • Identifying them is key to completely factoring polynomials.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the maximum or minimum value of the function. $$f(t)=-3+80 t-20 t^{2}$$

The effectiveness of a television commercial depends on how many times a viewer watches it. After some experiments an advertising agency found that if the effectiveness \(E\) is measured on a scale of 0 to \(10,\) then $$E(n)=\frac{2}{3} n-\frac{1}{90} n^{2}$$ where \(n\) is the number of times a viewer watches a given commercial. For a commercial to have maximum effectiveness, how many times should a viewer watch it?

A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost \(\$ 6\) and the society sells an average of 20 per week at a price of \(\$ 10\) each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it will lose 2 sales per week. (a) Find a function that models weekly profit in terms of price per feeder. (b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?

A highway engineer develops a formula to estimate the number of cars that can safely travel a particular highway at a given speed. She finds that the number \(N\) of cars that can pass a given point per minute is modeled by the function $$N(x)=\frac{88 x}{17+17\left(\frac{x}{20}\right)^{2}}$$ Graph the function in the viewing rectangle \([0,100]\) by \([0,60] .\) If the number of cars that pass by the given point is greater than \(40,\) at what range of speeds can the cars travel?

Graph the rational function \(f\), and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{3}+6 x^{2}-5}{x^{2}-2 x}, \quad g(x)=-x+4$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.