1.
Expand the polynomial \((2x-y)^6\text{.}\)
Section 1.2 The Basics
Subsection 1.2.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
Subsection 1.2.2 Introduction
In this activity, you will learn basic usage of some of the most common Maple commands:
-
expand() -
factor() -
simplify() -
plot()
Exercises 1.2.3 Exercises
2.
Factor the polynomial \(16x^4-160x^3y+600x^2y^2-1000xy^3+625y^4\text{.}\)
3.
Simplify the expression \(\dfrac{x^3-1}{x-1}\text{.}\)
4.
Now we would like Maple perform all three commands together.
Aside: Multi-line commands.
(a)
Have Maple expand the rational expression \(\dfrac{(x-y)^2+(x+y)^2}{x^3-y^3}\text{.}\)
(b)
Add a semicolon to the end of the line, followed by
simplify(%).
(c)
Add another semicolon to the end of the line, followed by
factor(%).
(d)
Hit Enter to run all three commands together.
You should see three outputs now: expanding, simplifying, and factoring.
5.
(Optional) Consider polynomials of the form \(x^p-1\text{,}\) where \(p\) is a prime number. Try factoring each of the following:
(a)
\(x^2 - 1\)
(b)
\(x^3 - 1\)
(c)
\(x^5 - 1\)
(d)
\(x^7 - 1\)
(e)
\(x^{19} - 1\)
Can you notice a pattern and show that these polynomials follow a particular form when factored? To explain the pattern, use the
button to create a new paragraph after the current line.
Aside: New paragraph shortcut.
6.
Plot the following two functions using separate
plot() commands and note the difference in domain:
(a)
\(x^{\sfrac{1}{3}}\)
(b)
\(surd(x,3)\)
State the difference in domain using a new paragraph.
Aside: The surd() command.
surd() command.
7.
On a new Maple input, create a plot of the following list of functions
\begin{gather*}
[ x^2, x^3, sqrt(x), surd(x,3), abs(x) ]
\end{gather*}
and include the following options (separated by commas).
-
x = -5..10(This specifies the \(x\)-axis) -
y = -5..10(This specifies the \(y\)-axis) -
colour = [red,blue,green,purple,orange]
Hint 1.
Hint 2.
An example of plotting multiple functions at once can be found in Section 8.3.
