/*! 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 57 Find all zeros of the polynomial... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 57

Find all zeros of the polynomial. \(P(x)=x^{4}-6 x^{3}+13 x^{2}-24 x+36\)

Short Answer

Expert verified
The zeros are the solutions to: (x-2)(x-3)^3 = 0.

Step by step solution

01

Apply the Rational Root Theorem

The Rational Root Theorem states that any rational solution (or zero) of the polynomial equation \(P(x)=a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_0=0\) is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\). For \(P(x)=x^{4}-6 x^{3}+13 x^{2}-24 x+36\), the possible rational roots are the factors of 36, which are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36\).

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.

Polynomial Zeros
Polynomial zeros are the values of the variable, usually denoted as x, which make the polynomial equal to zero. In other words, they are the solutions to the polynomial equation \(P(x) = 0\). These zeros are crucial in understanding the behavior of a polynomial function as they indicate where the graph of the polynomial will intersect the x-axis.

To find these zeros, various methods such as factoring, using the Rational Root Theorem, or applying synthetic division can be employed. Each root provides vital information which can further assist in graphing and comprehending the polynomial's nature.
Factor Theorem
The Factor Theorem is a powerful tool in finding the zeros of a polynomial. It states that a polynomial \(P(x)\) has a factor \((x - c)\) if and only if \(P(c) = 0\). This implies that \(c\) is a root of the polynomial.

In practical terms, if you can determine that substituting a number \(c\) for \(x\) in the polynomial results in zero, then \(x - c\) is indeed a factor of \(P(x)\). This connection between factors and roots is essential for breaking down more complicated polynomials into simpler, more manageable pieces.
  • If \(P(c) = 0\), then \(x - c\) is a factor.
  • This process helps simplify polynomials for easier solutions.
Polynomial Division
Polynomial division, specifically synthetic division, is a technique to divide a polynomial by another polynomial of lower degree. This method is particularly useful when working with the roots identified through the Factor Theorem.

Once a factor of the polynomial is identified, synthetic division allows us to simplify the polynomial by dividing it by this factor. The result is a new, simpler polynomial, which can then be analyzed to find additional zeros. This step is important as it constructs a bridge to easier polynomial factorization.
  • Streamline the process of finding additional roots.
  • Useful for reducing the degree of polynomials.
Factorization of Polynomials
Factorization of polynomials involves expressing a polynomial as a product of its factors. These factors can be linear, meaning they appear in the form \((x - r)\), or quadratic, involving higher degrees.

By factoring a polynomial, we break it down into simpler polynomials which reveal the roots more clearly. This process often follows the Rational Root Theorem and synthetic division as preliminary steps.
  • Makes solving polynomial equations straightforward.
  • Essential for understanding the structure and solutions of the polynomial.

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

Graph the rational function, and find all vertical asymptotes, x- and y-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$ r(x)=\frac{x^{4}-3 x^{3}+6}{x-3} $$

A box with a square base has length plus girth of 108 in. (Girth is the distance "around" the box.) What is the length of the box if its volume is 2200 in \(^{3} ?\)

Graph the rational function, and find all vertical asymptotes, x- and y-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$ r(x)=\frac{4+x^{2}-x^{4}}{x^{2}-1} $$

Drug Concentration After a certain drug is injected into a patient, the concentration \(c\) of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in minutes since the injection), the concentration (in mg/L) is given by $$ c(t)=\frac{30 t}{t^{2}+2} $$ (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of drug in the bloodstream?

Graph the rational function, and find all vertical asymptotes, x- and y-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$ y=\frac{x^{4}-3 x^{3}+x^{2}-3 x+3}{x^{2}-3 x} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.