Textbook Question
Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=−x3+x2+16x−16
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 53
Write a polynomial that represents the length of each rectangle. Transcription: The area of the rectangle is 0.5x3 - 0.3x2 + 0.22x + 0.06 square units and its width is x + 0.2 units
Verified step by step guidance1
Recall the formula for the area of a rectangle: \(\text{Area} = \text{Length} \times \text{Width}\).
Given the area as \(0.5x^{3} - 0.3x^{2} + 0.22x + 0.06\) and the width as \(x + 0.2\), set up the equation: \(0.5x^{3} - 0.3x^{2} + 0.22x + 0.06 = \text{Length} \times (x + 0.2)\).
To find the length, divide the area polynomial by the width polynomial: \(\text{Length} = \frac{0.5x^{3} - 0.3x^{2} + 0.22x + 0.06}{x + 0.2}\).
Perform polynomial division (either long division or synthetic division) to divide \(0.5x^{3} - 0.3x^{2} + 0.22x + 0.06\) by \(x + 0.2\).
The quotient from this division will be the polynomial expression representing the length of the rectangle.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Expressions
A polynomial is an algebraic expression consisting of terms with variables raised to non-negative integer powers and coefficients. Understanding how to manipulate polynomials, including addition, subtraction, multiplication, and division, is essential for working with expressions like the given area and width.
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Introduction to Algebraic Expressions
Area of a Rectangle
The area of a rectangle is found by multiplying its length by its width. Given the area and one dimension (width), you can find the other dimension (length) by dividing the area polynomial by the width polynomial.
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Polynomial Division
Polynomial division is the process of dividing one polynomial by another, similar to numerical long division. It is used here to find the length polynomial by dividing the area polynomial by the width polynomial.
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Related Practice
Textbook Question
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+4)(x−1)/(x+2)≤0
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Textbook Question
Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. 2x5+7x4−18x2−8x+8=0
Textbook Question
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+4)/x>0
Textbook Question
Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. h(x)=1/x2 − 4
Textbook Question
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. x/(x−3)>0