# Polynomials Quiz

Question

## Web resources available

This quiz tests the work covered in Lecture 5 and corresponds to the first half of Section 1.6 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).

There are further web quizzes at Wiley. Choose section 6 from this page.

Be aware that it doesn’t seem to accept the written answers so you will have to check whether your answers are correct when they print the correct answer. Question 15 didn’t make sense on 7/11/05

You may wish to start reviewing the section on powers at http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/zoo/powers.html and then continue to http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/zoo/polynomials.html, both pages have useful graphics.

You need to be able to expand and factorize quadratics so you may find http://home.xnet.com/ fidler/triton/math/review/mat085/factor/factor1.htm useful to review factoring.

There is a puzzle at http://www.univie.ac.at/future.media/moe/tests/fun1/erkennen.html which gets you to match quadratic and linear graphs to their formulae.

There are some java applets at http://id.mind.net/ zona/mmts/functionInstitute/polynomialFunctions/graphs/polynomialFunctionGraphs.html which let you play with the coefficients of polynomials of various degrees. This gives a very good idea of how the coefficients change the graph.

There is another function plotter at http://www.quickmath.com/ if you click on plot in equations. This site is very good so you may wish to explore other aspects as well.

## Question 1

Is the following statement true or false?
$1{0}^{6}{x}^{2}$ dominates ${2}^{x}$ as $x\to \infty \phantom{\rule{0.3em}{0ex}}.$
 a) True. b) False.

Not correct. Choice (a) is false.
Try again, an exponential function will always dominate a power function.
An exponential function will always dominate a power function.

## Question 2

Which of the following statements are correct? There may be more than one correct answer.
 a) $7{x}^{3}-2{x}^{2}+4x-1$ is a polynomial of degree 3 with leading coefficient 7. b) $7{x}^{3}-2{x}^{2}+4x-1$ is a polynomial of degree 7 with leading coefficient 3. c) $17+5{x}^{2}-12{x}^{3}-5{x}^{4}$ is a polynomial of degree 4 with leading coefficient 17. d) $17+5{x}^{2}-12{x}^{3}-5{x}^{4}$ is a polynomial of degree 4 with leading coefficient -5. e) $2{x}^{2}+3{x}^{-2}+1$ is not a polynomial. f) $2{x}^{2}+3{x}^{-2}+1$ is a polynomial of degree 2 with leading coefficient 2.

There is at least one mistake.
For example, choice (a) should be true.
The highest power of $x$ is 3 and this is degree of the polynomial and the coefficient of ${x}^{3}$ is 7 and this is the leading coefficient.
There is at least one mistake.
For example, choice (b) should be false.
Try again, read the definition of a polynomial on page 38.
There is at least one mistake.
For example, choice (c) should be false.
Try again, read the definition of a polynomial on page 38.
There is at least one mistake.
For example, choice (d) should be true.
The highest power of $x$ is 4 and this is degree of the polynomial and the coefficient of ${x}^{4}$ is -5 and this is the leading coefficient.
There is at least one mistake.
For example, choice (e) should be true.
In a polynomial the powers of $x$ must be positive.
There is at least one mistake.
For example, choice (f) should be false.
Try again, read the definition of a polynomial on page 38.
1. True. The highest power of $x$ is 3 and this is degree of the polynomial and the coefficient of ${x}^{3}$ is 7 and this is the leading coefficient.
2. False. Try again, read the definition of a polynomial on page 38.
3. False. Try again, read the definition of a polynomial on page 38.
4. True. The highest power of $x$ is 4 and this is degree of the polynomial and the coefficient of ${x}^{4}$ is -5 and this is the leading coefficient.
5. True. In a polynomial the powers of $x$ must be positive.
6. False. Try again, read the definition of a polynomial on page 38.

## Question 3

Consider the graph below, which gives a global view.

Which of the the following statements are correct?
 a) The minimum possible degree is 4 and the leading coefficient is negative. b) The minimum possible degree is 5 and the leading coefficient is negative. c) The minimum possible degree is 4 and the leading coefficient is positive. d) The minimum possible degree is 5 and the leading coefficient is positive.

Not correct. Choice (a) is false.
Try again, the maximum number of turns is one less than the minimum possible degree.
There are 4 turns so the minimum possible degree is 5. Since it is an odd power and the graph tends to $\infty$ as $x\to -\infty$ the leading coefficient is negative.
Not correct. Choice (c) is false.
Try again, the graph tends to $\infty$ as $x\to -\infty$ and there are 4 turns.
Not correct. Choice (d) is false.
Try again, the graph tends to $\infty$ as $x\to -\infty \phantom{\rule{0.3em}{0ex}}.$

## Question 4

Consider the 4 graphs below, which gives global views of the functions, and match them to possible formulae.
a)
 ${f}_{1}^{}\left(x\right)$ = $-2{x}^{3}+6{x}^{2}+2x-6$ ${f}_{2}^{}\left(x\right)$ = $-{x}^{2}+3x+4$ ${f}_{3}^{}\left(x\right)$ = $\frac{6}{16}{x}^{4}-6$ ${f}_{4}^{}\left(x\right)$ = $2{x}^{5}-4{x}^{4}-2x+4$
b)
 ${f}_{1}^{}\left(x\right)$ = $-2{x}^{3}+6{x}^{2}+2x-6$ ${f}_{2}^{}\left(x\right)$ = $2{x}^{5}-4{x}^{4}-2x+4$ ${f}_{3}^{}\left(x\right)$ = $\frac{6}{16}{x}^{4}-6$ ${f}_{4}^{}\left(x\right)$ = $-{x}^{2}+3x+4$
c)
 ${f}_{1}^{}\left(x\right)$ = $2{x}^{3}+6{x}^{2}+2x-6$ ${f}_{2}^{}\left(x\right)$ = $-{x}^{2}+3x+4$ ${f}_{3}^{}\left(x\right)$ = $\frac{6}{16}{x}^{4}-6$ ${f}_{4}^{}\left(x\right)$ = $-2{x}^{5}-4{x}^{4}-2x+4$
d)
 ${f}_{1}^{}\left(x\right)$ = $-2{x}^{3}+6{x}^{2}+2x-6$ ${f}_{2}^{}\left(x\right)$ = ${x}^{2}+3x+4$ ${f}_{3}^{}\left(x\right)$ = $-\frac{6}{16}{x}^{4}-6$ ${f}_{4}^{}\left(x\right)$ = $2{x}^{5}-4{x}^{4}-2x+4$

Try again, ${f}_{2}^{}$ has to be of even degree and ${f}_{4}^{}$ has to be of odd degree.