Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

27.2: Review of polynomials and rational functions

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 27.2.1

Divide the polynomials: 2x3+x29x82x+3

Answer

x2x3+12x+3

Exercise 27.2.2

Find the remainder when dividing x3+3x25x+7 by x+2.

Answer

21

Exercise 27.2.3

Which of the following is a factor of x4002x99+1: x1,x+1,x0

Answer

x1 is a factor, x+1 is not a factor, x0 is not a factor

Exercise 27.2.4

Identify the polynomial with its graph.

  1. clipboard_e0c0295236f728781dd9f837be2a680f2.png
  2. clipboard_ea73579cf5c91f70adf630ab2ee0b3612.png
  3. clipboard_e86c1d73b604adf1df9c67be8ac6bb649.png
  4. clipboard_e1b082903b9681e6702d6ec155312e9db.png
  1. f(x)=x2+2x+1 graph: _______________
  2. f(x)=x3+3x23x+2 graph: _______________
  3. f(x)=x33x2+3x+1 graph: _______________
  4. f(x)=x44x3+6x24x+2 graph: _______________
Answer
  1.  iii) 
  2.  iv) 
  3. i)
  4.  ii) 

Exercise 27.2.5

Sketch the graph of the function: f(x)=x410x30.01x2+0.1x

  • What is your viewing window?
  • Find all roots, all maxima and all minima of the graph with the calculator.
Answer

clipboard_e204f31b4ec80b0feba7e396e7b6b5199.png

Exercise 27.2.6

Find all roots of f(x)=x3+6x2+5x12.

Use this information to factor f(x) completely.

Answer

f(x)=(x1)(x+3)(x+4)

Exercise 27.2.7

Find a polynomial of degree 3 whose roots are 0, 1, and 3, and so that f(2)=10.

Answer

f(x)=(5)x(x1)(x3)

Exercise 27.2.8

Find a polynomial of degree 4 with real coefficients, whose roots include 2, 5, and 32i.

Answer

f(x)=(x+2)(x5)(x(32i))(x(3+2i)) (other correct answers are possible, depending on the choice of the first coefficient)

Exercise 27.2.9

Let f(x)=3x212x22x3. Sketch the graph of f. Include all vertical and horizontal asymptotes, all holes, and all x- and y-intercepts.

Answer

f(x)=3(x2)(x+2)(x3)(x+1) has domain D=R{1,3}, horizontal asympt. y=3, vertical asympt. x=1 and x=3, no removable discont., x-intercepts at x=2 and x=2 and x=3, y-intercept at y=4, graph:

clipboard_e28cf3f49f7fcd7d03fb0b4fd6823fa75.png

Exercise 27.2.10

Solve for x:

  1. x4+2x<2x3+x2
  2. x2+3x7
  3. x+1x+42
Answer
  1. (1,0)(1,2)
  2. (,3372][3+372,)
  3. (,7](4,)

This page titled 27.2: Review of polynomials and rational functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?