/*! 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 61 Find \(k\) so that \(4 x+3\) is ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 61

Find \(k\) so that \(4 x+3\) is a factor of $$20 x^{3}+23 x^{2}-10 x+k$$

Short Answer

Expert verified
The value of \(k\) that makes \(4x+3\) a factor of \(20 x^{3}+23x^{2}-10x+k\) is -13.25. This value is obtained by equating the polynomial to zero for the value of \(x\) that nullifies the factor, and then solving for \(k\).

Step by step solution

01

Solve \(4x+3\) for \(x\)

First, solve the equation \(4x+3=0\) for \(x\) to obtain \(x=-3/4\). Here's the calculation: \(4x+3=0\) Subtraction 3 from both sides gives: \(4x=-3\) Dividing both sides by 4 gives: \(x=-3/4\)
02

Plug \(x=-3/4\) into the polynomial

Next, you need to plug \(x=-3/4\) into the polynomial \(20 x^{3}+23x^{2}-10x+k\) and equate it to zero, since a factor should nullify the polynomial. \(20(-3/4)^3+23(-3/4)^2-10(-3/4)+k=0\)
03

Compute and solve for \(k\)

Simplify the equation to find the value of \(k\) After mathematical manipulation, the equation simplifies to: \(-27+131/4+15/2+k=0\) Simplify further to: \(-27+32.75+7.5+k=0\) Which simplifies to: \(k= -13.25\) \ k is equal to negative 13.25

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

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) 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.

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does how do you find its equation?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-3(x-1)^{2}\left(x^{2}-4\right)$$

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+7 x^{2}-4 x-28$$

The heat generated by a stove element varies directly as the square of the voltage and inversely as the resistance. If the voltage remains constant, what needs to be done to triple the amount of heat generated?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.