/*! 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 43 Solve the equation \(2 x^{3}-5 x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 43

Solve the equation \(2 x^{3}-5 x^{2}+x+2=0\) given that 2 is a zero of \(f(x)=2 x^{3}-5 x^{2}+x+2\)

Short Answer

Expert verified
The solution to the equation \(2 x^{3}-5 x^{2}+x+2=0\) are \(x=2, 2, \frac{1}{4}\).

Step by step solution

01

Confirm the Given Zero

The first step is to confirm that the given number is a zero of the function. This is done by substituting the given number into the equation to check if it equals zero. So we substitute \(x=2\) into the equation \(f(x)=2x^{3}-5x^{2}+x+2\) and if \(f(2)=0\), then indeed 2 is a zero of the equation. Checking this in detail leads to \(f(2)=2(2^{3})-5(2^{2})+(2)+2 = 0\). Therefore, 2 is a zero of the equation.
02

Perform Synthetic Division

Perform synthetic division, which is a shorthand version of polynomial division, using the given zero to reduce the equation to a quadratic. The process of synthetic division is performed as follows: List the coefficients of the equation (2, -5, 1, and 2), then write down the given zero. Bring the first coefficient directly down and multiply it by the given zero, then add this to the next coefficient. Repeat these operations until the end of the coefficients' list. The final number should be zero if the given zero is indeed a root. Doing this, we get \(2x^{2}-9x+4=0\).
03

Solve the Quadratic Equation

We obtain a quadratic equation from the synthetic division which we can solve It using the quadratic formula \(x = \frac{-b 卤 \sqrt{b^{2}-4ac}}{2a}\), where \(a\), \(b\) and \(c\) are coefficients of the equation. In our quadratic equation \(a=2\), \(b=-9\), \(c=4\). Plugging these values into the formula, we obtain \(x=\frac{9\pm\sqrt{81-32}}{4}\). Simplifying, this gives \(x=2, \frac{1}{4}\). Thus, the three solutions of the cubic equation are \(2, 2, \frac{1}{4}\).

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

Body-mass index, or BMI, takes both weight and height into account when assessing whether an individual is underweight or overweight. BMI varies directly as one's weight, in pounds, and inversely as the square of one's height, in inches. In adults, normal values for the BMI are between 20 and \(25,\) inclusive. Values below 20 indicate that an individual is underweight and values above 30 indicate that an individual is obese. A person who weighs 180 pounds and is 5 feet, or 60 inches, tall has a BMI of \(35.15 .\) What is the BMI, to the nearest tenth, for a 170 -pound person who is 5 feet 10 inches tall? Is this person overweight?

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies jointly as \(m\) and the square of \(n\) and inversely as \(p\) \(y=15\) when \(m=2, n=1,\) and \(p=6 .\) Find \(y\) when \(m=3, n=4,\) and \(p=10\).

Kinetic energy varies jointly as the mass and the square of the velocity. A mass of 8 grams and velocity of 3 centimeters per second has a kinetic energy of 36 ergs. Find the kinetic energy for a mass of 4 grams and velocity of 6 centimeters per second.

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degreeof \(p\) and \(q\) are as small as possible. More than one correct finction may be possible. Graph your function using a graphing utility to verify that it has the required propertics \(f\) has vertical asymptotes given by \(x=-2\) and \(x=2\) a horizontal asymptote \(y=2, y\) -intercept at \(\frac{9}{2}, x\) -intercepts at \(-3\) and \(3,\) and \(y\) -axis symmetry.

Write an equation that expresses each relationship. Then solve the equation for \(y .\) \(x\) varies directly as the cube root of \(z\) and inversely as \(y .\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.