Master Polynomial Concepts with Comprehensive Resources

Find the sum of each arithmetic series
${\displaystyle \sum _{p=4}^{11}(2p+1)}$.

Find the sum of each arithmetic series
${\displaystyle \sum _{j=1}^5(9-4j)}$

To find: The sum of the given arithmetic sequence.
The sequence ${\displaystyle 2\sum n=1802n-5}$

Sequences and Series
1.[4.K] Is each sequence arithmetic, geometric or neither? State the general term (simplify fully).
a)81,75,69,...
b)2187,729,243,...
2.[2,K] Determine the first four terms of the sequence if the general term is${\displaystyle t_n=\frac{5n+2}{2n+1}}$.

Insert four arithmetic means between 20 and -10.

Find the sum of each arithmetic series
${\displaystyle \sum _{k=3}^8(5k-10)}$

Find S for each arithmetic series described
${\displaystyle S_{13};a_1=16,a_n=98}$

Find the indicated term in each arithmetic sequence
${\displaystyle a_7;a_1=-4;d=7}$

Find the sum of each arithmetic series
${\displaystyle 5,7,9\cdots 27}$

The recursive forula of an arithmetic sequence is described below.
${\displaystyle a_1=19,a_n-1+6}$
Which represent the explicit of this aritmetic sequence?
A. ${\displaystyle a_n=6n+13}$
B. ${\displaystyle a_n=6n+19}$
C. ${\displaystyle a_n=13n+6}$
D. ${\displaystyle a_n=19n+6}$

Insert three arithmetic means between 3 and -5.

Find the sum of the finite arithmetic series.
${\displaystyle 1+5+9+\cdots +157}$

Find S for each arithmetic deries described
${\displaystyle 21.S_{16};d=\frac{-2}{3},a_n=44}$

Find the indicated term in each arithmetic sequence
${\displaystyle a_{19}}$ for ${\displaystyle 1,\frac{7}{3},\frac{11}{3}}$

Insert three arithmetic means between
-15 and -4.

This problem deals with polynomials in ${\displaystyle Z_2\left[X\right]}$. Remember these have ${\displaystyle 0-1}$ coefficients. You use the usual algorithms, but with all arithmetic done mod 2.
a) It was observed in class that the constant term of an irreducible polynomial is nonzero. Show further that the Hamming weight (number of nonzero coeffs) of an irreducible polynomial of degree ${\displaystyle ⟩1}$ is odd. [Hint: Use the remainder theorem.]
b) Only one degree 2 polynomial f meets the conditions of a). What is it? Give the addition and multiplication tables for polynomial arithmetic mod f.
c) Using similar ideas, find all the irreducibles of degrees 3 and 4. (Note: For degree 4, you will need to consider a possible quadratic factor.)

Determine the multiplicative inverse of ${\displaystyle (x^3+x+1)}$ in GF${\displaystyle \left(2^4\right)}$ with ${\displaystyle m\left(x\right)=x^4+x+1}$.

The scoring of modern IQ tests is such that Intelligence Quotients (IQs) have Normal distribution with ${\displaystyle \mu =100}$ and ${\displaystyle \sigma =15}$
a) What percent of people have IQ less than 80?
b) What percent of people have IQ greater than 120?
c) Mensa International is a non-profit organization that accepts only people with IQ within the top 2%.
What level of IQ qualifies one to be a member of Mensa?

Using modular polynomial arithmetic, perform the following multiplication over ${\displaystyle GF\left(2^3\right)}$ with the irreducible polynomial ${\displaystyle x^3+x+1}$.
Convert the final result to decimal.
${\displaystyle 111\times 110}$

For polynomial arithmetic with coefficients in ${\displaystyle Z_{10}}$, perform the following calculations.
a.${\displaystyle (7x+2)-(x^2+5)}$
b.${\displaystyle (6x^2+x+3)\times (5x^2+2)}$