/*! 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 19 Find all solutions of the equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 19

Find all solutions of the equation. $$x^{3}-x^{2}-10 x-8=0$$

Short Answer

Expert verified
The solutions are \(x = -1\), \(x = 4\), and \(x = -2\).

Step by step solution

01

Use the Rational Root Theorem

The Rational Root Theorem states that any rational solution of the polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0\) with integer coefficients 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 the given polynomial \(x^3-x^2-10x-8=0\), \(a_0 = -8\) and \(a_n = 1\). Therefore, the possible rational roots are the factors of -8: \(\pm 1, \pm 2, \pm 4, \pm 8\).
02

Test the Possible Rational Roots

Evaluate the polynomial at each of the possible rational roots. Start with \(x = 1\):\[f(1) = 1^3 - 1^2 - 10 imes 1 - 8 = 1 - 1 - 10 - 8 = -18\]Try \(x = 2\):\[f(2) = 2^3 - 2^2 - 10 imes 2 - 8 = 8 - 4 - 20 - 8 = -24\]Next, try \(x = -1\):\[f(-1) = (-1)^3 - (-1)^2 - 10 imes (-1) - 8 = -1 - 1 + 10 - 8 = 0\]Since \(f(-1) = 0\), \(x = -1\) is a root.
03

Factor the Polynomial

Since \(x = -1\) is a root, \((x + 1)\) is a factor of \(x^3-x^2-10x-8\). Use synthetic division to divide the polynomial by \(x + 1\), which gives:\(\begin{array}{r|rrr}-1 & 1 & -1 & -10 & -8 \ & & -1 & 2 & 8 \\hline & 1 & -2 & -8 & 0 \\end{array}\)This results in the quotient \(x^2 - 2x - 8\).
04

Solve the Quadratic Equation

Now, solve the quadratic equation \(x^2 - 2x - 8 = 0\) by factoring. Look for two numbers that multiply to -8 and add to -2:\((x - 4)(x + 2) = 0\).Thus, the roots are \(x = 4\) and \(x = -2\).
05

List All Solutions

The solutions to the original equation \(x^3-x^2-10x-8=0\) are \(x = -1\), \(x = 4\), and \(x = -2\).

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.

Rational Root Theorem
The Rational Root Theorem is a helpful tool when solving polynomial equations. It offers a systematic method to find possible rational roots of a polynomial.
  • The theorem states that any rational root of a polynomial with integer coefficients will have the form \(\frac{p}{q}\). Here, \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.
  • For example, in the polynomial \(x^3 - x^2 - 10x - 8 = 0\), the constant term is \(-8\) and the leading coefficient is \(1\).
Since the leading coefficient is \(1\), any rational root must simply be a factor of \(-8\). These possible roots are \( \pm 1, \pm 2, \pm 4, \pm 8\). By testing these roots in the polynomial, one can identify which values actually resolve the equation, simplifying the process of finding all solutions.
Synthetic Division
In polynomial equations, synthetic division is a simplified method of dividing a polynomial by a linear factor. It is much quicker and more manageable than long division.
  • Using synthetic division begins once you've identified a root, say \(x = -1\).
  • You then rewrite the division process using the coefficients of the polynomial. Here, the coefficients from \(x^3-x^2-10x-8\) are \(1, -1, -10, -8\).
To perform synthetic division:1. List the coefficients \(1, -1, -10, -8\).2. Use the root '\(-1\)' to divide through these coefficients.3. Execute the division to determine the simplified polynomial.This results in \(x^2 - 2x - 8\). Synthetic division proves efficient, especially for polynomials of higher degrees or when manually solving equations.
Quadratic Equations
After using synthetic division, we often obtain a quadratic equation, which is a simpler form to solve. Quadratic equations are of the form \(ax^2 + bx + c = 0\). Our divided polynomial resulted in \(x^2 - 2x - 8 = 0\). Common methods to solve quadratic equations include:
  • Factoring: Look for two numbers that multiply to the constant term \(c\) and add up to the coefficient \(b\). In our example, it's \(-8\) and \(-2\), leading to roots \(x = -2\) and \(x = 4\).
  • Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is another method, though not needed here.
  • Completing the Square: Another way to solve, especially useful for more complex expressions.
Sometimes, factoring directly leads to the solution, minimizing calculation errors and saving time!
Factoring Polynomials
Factoring polynomials involves breaking down a complex polynomial into a product of simpler polynomials. This is a crucial step in solving polynomial equations because it often reveals the roots quite easily. For any polynomial, consider:
  • Finding factors of the constant term and leading coefficients.
  • Testing potential roots by substituting them into the polynomial.
In our case, we identified \((x + 1)\) as a factor after finding \(x = -1\) was a root. After dividing by \((x + 1)\), the quotient \(x^2 - 2x - 8\) was factored further into \((x - 4)(x + 2)\), revealing more roots. Factoring transforms complex problems into solvable parts, making it a powerful technique in algebraic problem-solving.

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

If \(n\) is an odd positive integer, prove that a polynomial of degree \(n\) with real coefficients has at least one real zero.

The population density \(D\) (in people \(/ \mathrm{mi}^{2}\) ) in a large city is related to the distance \(x\) (in miles) from the center of the city by $$D=\frac{5000 x}{x^{2}+36}$$ (a) What happens to the density as the distance from the center of the city changes from 20 miles to 25 miles? (b) What eventually happens to the density? (c) In what areas of the city does the population density exceed 400 people/mi \(^{2}\) ?

Use synthetic division to show that \(c\) is a zero of \(f(x)\). $$f(x)=27 x^{4}-9 x^{3}+3 x^{2}+6 x+1 ; \quad c=-\frac{1}{3}$$

A solid wood sphere whose density is less than that of water will float. The depth \(d\) that the sphere will sink into the water is determined by the equation $$\frac{4 k}{3} \pi r^{3}-\pi d^{2} r+\frac{1}{3} \pi d^{3}=0$$ where \(r\) is the radius of the sphere and \(k\) is a positive constant less than or equal to \(1 .\) If \(r=6 \mathrm{cm},\) graphically estimate \(d\) for each constant \(k\) Pine sphere in water \(k=0.7\)

Simplify \(f(x),\) and sketch the graph of \(f\). $$f(x)=\frac{x^{2}+4 x+4}{x^{2}+3 x+2}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.