Here’s a short homework assignment on proof by indication, to be collected on Monday, February 6th.
Show that \(6^n+4\) is divisible by \(10\) for all \(n \in \mathbb{Z}^+\).
You can download the template from class here.
The Binomial Theorem
Find the simplified expansion of \((a+b)^7\). You can use any method you prefer, but the Binomial Theorem may save you some work!
Sequences and Series Test
We’ll have our test on Sequences and Series on Tuesday, January 17th.
In order to prepare for the test, complete the following questions.
Pages 200–205 questions 1–3, 5, 6, 9, 13, 19, 23, 25, 27, 29, 35
Infinite Geometric Series
For tomorrow’s lesson, try to complete page 173 questions 5, 6, 23, 27, and 28.
Series Break Homework
Complete the following questions during the break. Remember, if you get stuck you can leave a comment below. If you see a question in the comments and you have a suggestion to make, post that too!
Pages 172–173 questions 1, 2, 3, 11, 12, and 20
Additionally, see if you can complete all of the question below.
Consider the series shown here.
\[\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+\cdots\]
- Find the formula for \(S_n\)
- Write down the value of
- \(S_3\)
- \(S_{10}\)
- \(S_{100}\)
- What do you notice happening to the value of \(S_n\) as \(n\) increases? Can you use the formula for \(S_n\) to explain why this is happening?
Polynomials Test
We’ve now covered all the material that will appear on our next test, to be held on Monday, November 28th.
The following questions from our textbook will help you prepare for the test.
Page 150 questions 11–21, 28
Inequalities
Here’s a question to consider tonight. This question can be answer in much the same way as the inequalities we considered in class today (there is one minor difference—see if you can figure this out!).
Solve the inequality \(\displaystyle{\frac{x+1}{x-4}\leq \frac{1}{x-2}}\).
Factors, Roots, and the Fundamental Theorem of Algebra
We’ve now covered a number of important theorems concerning polynomials. For our next lesson complete the questions below.
Complete pages 124–125, questions 6, 11, 18, 22, 24, 26, 27, 32, 33, 34
In addition to these questions, if you’re ready for a challenge, try to answer any of the questions shown in the Polynomials Super Challenge. These are not assigned as homework, but if you can correctly complete question 1 you’ll get 1 bonus mark on our next test, and if you can correctly complete question 2 you’ll get 2 bonus marks on our next test. You have until the date of our next test to complete any of these challenge questions. Good luck!
Working with the Factor Theorem
Below is the question we considered at the end of our lesson today. Complete this question for our next lesson.
Let \(p(x)=2x^3+ax^2-29x+60\). The polynomial \(p\) is divisible by \((x-3)\).
a) Find \(a\).
b) \(-4\) is also a zero of \(p\). Find the third zero of \(p\).
The Remainder Theorem
Complete the following questions for tomorrow’s lesson.
Page 125 questions 28 and 29