/*! 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 37 Find the zeros of each polynomia... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 37

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=x^{3}+3 x^{2}-6 x-8$$

Short Answer

Expert verified
The zeros of the polynomial \(P(x) = x^{3}+3x^{2}-6x-8\) are \(x = -2, x = -1, x = -4\). Each root has multiplicity 1.

Step by step solution

01

Identify the polynomial

The polynomial given in the exercise is \(P(x) = x^{3}+3x^{2}-6x-8\). This is a cubic polynomial. The aim is to find the roots (zeros) of this polynomial.
02

Try to factor the polynomial

Start with integer roots if available, using the Rational Root Theorem. From the Rational Root Theorem, possible rational zeros are factors of the constant term divided by factors of the leading coefficient. For this polynomial, the possible rational roots are \(\pm 1, \pm 2, \pm 4, \pm 8\). Substitute these possible zeros into \(P(x)\), and check if \(P(x) = 0\). It is found that \(x = -2\) and \(x = -1\) are zeros of the polynomial.
03

Apply synthetic division

Use synthetic division to find other roots. When \(x = -2\) is used, the resulting polynomial is \(x^2+x+4\). Now use \(x = -1\) to divide this further, the remaining polynomial is \(x+4\). Thus, the roots are \(x = -2, x = -1, x = -4\).
04

Identify multiplicities

Each root appears only once in the factorization, so each root has multiplicity 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!

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

INSCRIBED QUADRILATERAL Isaac Newton discovered that if a quadrilateral with sides of lengths \(a, b\) \(c,\) and \(x\) is inscribed in a semicircle with diameter \(x\) then the lengths of the sides are related by the following equation. $$x^{3}-\left(a^{2}+b^{2}+c^{2}\right) x-2 a b c=0$$ Given \(a=6, b=5,\) and \(c=4,\) find \(x .\) Round to the nearest hundredth.

Use Descartes' Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. $$P(x)=6 x^{4}+23 x^{3}+19 x^{2}-8 x-4$$

Find a polynomial function \(P(x)\) that has the indicated zeros. Zeros: \(4+3 i, 5-i ;\) degree 4

COST CUTTING At the present time, a nutrition bar in the shape of a rectangular solid measures 0.75 inch by 1 inch by 5 inches. To reduce costs the manufacturer has decided to decrease each of the dimensions of the nutrition bar by \(x\) inches. What value of \(x,\) rounded to the nearest thousandth of an inch, will produce a new nutrition bar with a volume that is 0.75 cubic inch less than the present bar's volume?

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=3 x^{6}-10 x^{5}-29 x^{4}+34 x^{3}+50 x^{2}-24 x-24$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

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.