/*! 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 11 Find a polynomial function of de... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 11

Find a polynomial function of degree 3 with the given numbers as zeros. $$-\frac{1}{3}, 0,2$$

Short Answer

Expert verified
y = 3x^3 - 5x^2 - 2x

Step by step solution

01

Write the Zeros as Factors

The polynomial function of degree 3 will be in the form of y = k(x - r_1)(x - r_2)(x - r_3),where k is a constant, and r_1, r_2, and r_3 are the given zeros. For the zeros $$-\frac{1}{3}, 0, 2$$, we have y = k(x + \frac{1}{3})(x)(x - 2).
02

Simplify the Factors

Since having a fractional zero can make the polynomial less clear, multiply the factor $$\frac{1}{3}$$ by 3, to get integer coefficients. Therefore,y = k(3x + 1)(x)(x - 2).
03

Multiple the Factors

First, multiply the factors (x) and (3x + 1):y = k [(x)(3x + 1)](x - 2)y = k (3x^2 + x)(x - 2).
04

Expand the Polynomial

Next, distribute the (3x^2 + x) term over the (x - 2) term:y = k [(3x^2 + x)(x - 2)]y = k (3x^3 - 6x^2 + x^2 - 2x)y = k (3x^3 - 5x^2 -2x).
05

Determine the Leading Coefficient

Since the problem doesn't specify a leading coefficient, we can choose k = 1 for simplicity. Therefore, the final polynomial isy = 3x^3 - 5x^2 - 2x.

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.

Degree of a Polynomial
In a polynomial function, the degree is the highest power of the variable, typically denoted as x. For example, in the polynomial \(3x^3 - 5x^2 - 2x\), the highest power of x is 3, which means this is a polynomial of degree 3. Understanding the degree of a polynomial helps us to determine the shape of its graph and the possible number of zeros it can have. A polynomial of degree n can have at most n real zeros.
Zeros of a Polynomial
Zeros of a polynomial (or roots) are the values of x that make the polynomial equal to zero. For example, if we have a polynomial with zeros at \(-\frac{1}{3}\), 0, and 2, it means that these values when substituted for x will make the polynomial zero. Zeros play a crucial role in factoring and graphing polynomials. To find the polynomial when zeros are given, we convert them into factors. Thus, zeros at \(-\frac{1}{3}\), 0, and 2 correspond to factors of \((x + \frac{1}{3})\), (x), and \((x - 2)\) respectively.
Factoring Polynomials
Factoring a polynomial involves writing it as a product of its factors. When given the zeros of a polynomial, like \(-\frac{1}{3}\), 0, and 2, we can turn these into factors. For simplicity, we often adjust factors to avoid fractions. For example, the factor corresponding to \(-\frac{1}{3}\) can be multiplied by 3, resulting in \(3x + 1\). The polynomial can then be expressed as a product of these factors: \((3x + 1)(x)(x - 2)\). This process simplifies solving and graphing the polynomial.
Polynomial Expansion
Polynomial expansion involves multiplying the factors to get the polynomial in standard form. For example, to expand \(y = (3x + 1)(x)(x - 2)\), we start by multiplying two of the factors: \((3x + 1)(x) = 3x^2 + x\). Next, we distribute this result over the third factor: \[ (3x^2 + x)(x - 2) = 3x^3 - 6x^2 + x^2 - 2x = 3x^3 - 5x^2 - 2x\].\ Note how each term is multiplied systematically. The final expanded polynomial is \(3x^3 - 5x^2 - 2x\), which is in standard form and ready for further analysis such as graphing or finding more properties.

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

Fill in the blank with the correct term. Some of the given choices will not be used. Others will be used more than once. $$\begin{array}{ll}x \text { -intercept } & \text { midpoint formula } \\ y \text { -intercept } & \text { horizontal lines } \\ \text { odd function } & \text { vertical lines } \\ \text { even function } & \text { point-slope equation } \\ \text { domain } & \text { slope-intercept equation } \\ \text { range } & \text { difference quotient } \\ \text { slope } & f(x)=f(-x) \\\ \text { distance formula } & f(-x)=-f(x)\end{array}$$ A(n) ______ is a point \((a, 0)\)

List the critical values of the related function. Then solve the inequality. $$\frac{2}{x^{2}-4 x+3} \leq \frac{5}{x^{2}-9}$$

Find the zeros of the function. $$g(x)=-x^{3}-2 x^{2}[4.1]$$

List the critical values of the related function. Then solve the inequality. $$\frac{-4}{2 x+5} < 0$$

Beam Deflection. A beam rests at two points \(A\) and \(B\) and has a concentrated load applied to its center. Let \(y=\) the deflection, in feet, of the beam at a distance of \(x\) feet from \(A .\) Under certain conditions, this deflection is given by $$y=\frac{1}{13} x^{3}-\frac{1}{14} x$$ Find the zeros of the polynomial on the interval \([0,2]\) GRAPH CANT COPY
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.