/*! 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 23 Show that the number is a zero o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 23

Show that the number is a zero of \(f(x)\) of the given multiplicity, and express \(f(x)\) as a product of linear factors. $$\begin{aligned} f(x)=x^{4}+7 x^{3}+13 x^{2}-3 x-18 & \\ &-3 \text { (multiplicity 2) } \end{aligned}$$

Short Answer

Expert verified
\(f(x) = (x + 3)^2 (x^2 + 4x - 2)\) with \(-3\) having multiplicity 2.

Step by step solution

01

Plug in the Given Zero

The first step is to verify that \(-3\) is indeed a zero of the function \(f(x)\). To do this, substitute \(x = -3\) into \(f(x)\):\[f(-3) = (-3)^4 + 7(-3)^3 + 13(-3)^2 - 3(-3) - 18\] Calculate the value to ensure \(f(-3) = 0\).
02

Perform Synthetic Division

Since \(x = -3\) is a zero of multiplicity 2, it implies \(x + 3\) is a factor. Perform synthetic division of \(f(x)\) by \(x + 3\) twice to verify the multiplicity and reduce \(f(x)\). First division gives us a quotient polynomial. Repeat the division with the quotient until we reach a quadratic expression.
03

Verify Multiplicity and Factor Further

After performing synthetic division twice, verify the multiplicity by ensuring each division results in zero remainder. The resulting quotient should be a quadratic polynomial if \(-3\) is truly a zero of multiplicity 2.
04

Express as Product of Linear Factors

Factor the resulting quadratic polynomial (from Step 2) into linear factors, if possible. The complete expression for \(f(x)\) as a product of linear factors should include the factor \((x + 3)^2\) to show the zero of multiplicity 2.
05

Write Final Expression

Combining all the factors, write \(f(x)\) as a product of linear factors:\[f(x) = (x + 3)^2 (x^2 + 4x - 2)\]If the quadratic can further be factored into linear factors, do so.

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.

Synthetic Division
Synthetic division is a streamlined way of dividing polynomials, especially useful when you want to test potential zeros of a function.
Unlike long division, synthetic division is faster and involves fewer steps. It's usually applied when the divisor is a linear factor of the form \( x - c \).
  • First, list the coefficients of the polynomial you're dividing.
  • The number you're dividing by must be of opposite sign to what appears in the factor.
  • The process involves bringing down the first coefficient, multiplying it by the divisor, adding to the next coefficient, and repeating.
For the given polynomial \( f(x) = x^4 + 7x^3 + 13x^2 - 3x - 18 \) and zero \(-3\), perform synthetic division by \(x + 3\).
This will confirm \(-3\) is a zero.
Then, repeat the process to address the multiplicity.
Multiplicity of Zeros
Multiplicity refers to the number of times a particular zero appears in a polynomial. A zero of multiplicity 2, for example, implies the zero appears twice.
Multiplicities affect the function's graph.
For instance, a zero with even multiplicity will touch the x-axis but not cross it. Conversely, an odd multiplicity zero will cross the x-axis.
  • To verify a zero's multiplicity, perform synthetic division. Multiple successful divisions with zero remainders signify the multiplicity.
  • In this problem, \(-3\) is stated as a zero with multiplicity 2, meaning we can divide \(f(x)\) by \((x + 3)^2\).
  • Each division reduces the degree of the polynomial by 1, thus after two divisions, the polynomial degree reduces by 2.
When dividing twice by \((x + 3)\), the remainder being zero both times verifies the zero's multiplicity.
Polynomial Factoring
Factoring polynomials involves expressing them as a product of linear factors, which can often be used to find zeros or roots.
The goal in this exercise is to break down \(f(x)\) completely by factoring out known elements.
  • After dividing the polynomial using synthetic division, the remaining polynomial is a quadratic.
  • The known zero multiplies into the quadratic, aiding further factoring.
For \(f(x) = (x + 3)^2(x^2 + 4x - 2)\), first establish the multiplicity zero factor as \((x + 3)^2\).
Then, look to factor \(x^2 + 4x - 2\):
Find two numbers that multiply to \(-2\) and add to \(4\).
If no numbers fit, use the quadratic formula to find potential linear factors as roots.
Thus, complete the expression as a product of linear factors.

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

For a particular salmon population, the relationship between the number \(S\) of spawners and the number \(R\) of offspring that survive to maturity is given by the formula $$R=\frac{4500 \mathrm{s}}{s+500}$$ (a) Under what conditions is \(R>S ?\) (b) Find the number of spawners that would yield \(90 \%\) of the greatest possible number of offspring that survive to maturity. (c) Work part (b) with 80 \% replacing 90 \(\$ 6 dollar. (d) Compare the results for \)S\( and \)R$ (in terms of percentage increases) from parts (b) and (c).

Is there a polynomial of the given degree \(n\) whose graph contains the indicated points? $$\begin{aligned} &n=4\\\ &(1.25,0),(2,0),(2.5,56.25),(3,128.625),(6.5,0)\\\ &(9,-307.75),(10,0) \end{aligned}$$

Grade point average (a) A student has finished 48 credit hours with a GPA of 2.75. How many additional credit hours \(y\) at 4.0 will raise the student's GPA to some desired value \(x ?\) (Determine \(y\) as a function of \(x\).) (b) Create a table of values for \(x\) and \(y,\) starting with \(x=2.8\) and using increments of 0.2 (c) Graph the function in part (a) in the viewing rectangle \([2,4]\) by \([0,1000,100]\) (d) What is the vertical asymptote of the graph in part (c)? (e) Explain the practical significance of the value \(x=4\)

Because of the combustion of fossil fuels, the concentration of carbon dioxide in the atmosphere is increasing. Research indicates that this will result in a greenhouse effect that will change the average global surface temperature. Assuming a vigorous expansion of coal use, the future amount \(A(t)\) of atmospheric carbon dioxide concentration can be approximated (in parts per million) by $$ A(t)=-\frac{1}{2400} t^{3}+\frac{1}{20} t^{2}+\frac{7}{6} t+340 $$ where \(t\) is in years, \(t=0\) corresponds to \(1980,\) and \(0 \leq t \leq 60 .\) Use the graph of \(A\) to estimate the year when the carbon dioxide concentration will be 400 .

Find an equation of a rational function \(f\) that satisfles the given conditions. vertical asymptote: \(x=4\) horizontal asymptote: \(y=-1\) \(x\) -intercept: 3
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

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.