Master Polynomial Concepts with Comprehensive Resources

Expand Q to prove that the polynomials P and Q are the same.
${\displaystyle P\left(x\right)=3x^4-5x^3+x^2-3x+5}$
${\displaystyle Q\left(x\right)=(((3x-5)x+1)x-3)x+5}$
Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial ${\displaystyle R\left(x\right)=x^5-2x^4+3x^3-2x^2+3x+4}$ in "nested" form, like the polynomial Q. Use the nested form to find R(3) in your head.
Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value of a polynomial using synthetic division?

What is the derivative of the function
${\displaystyle f\left(x\right)=\frac{3x-5}{5x+2}}$?

Find a polynomial equation with real coefficients that has the given roots.
${\displaystyle -1,4-2i}$

In this problem we are working over the field ${\displaystyle Z_7}$, the integers mod 7. All arithmetic is done mod 7. First, write out the addition and multiplication tables for ${\displaystyle Z_7}$. Secondly, we want codewords to be sequences of elements from ${\displaystyle Z_7}$. Suppose that in order to detect and correct a possible error we will add two additional digits to the sequence. In this example, we have a sequence of length 3 that we wish to transmit, call it y1, y2, y3, so we find a polynomial f of degree at most 2 for which ${\displaystyle f\left(1\right)=y1,f\left(2\right)=y2,f\left(3\right)=y3}$ (all of this is done over ${\displaystyle Z_7}$.) (To find f we use the 6 functions I discussed in class, but again the arithmetic is done mod 7.) To detect and correct a possible error we compute ${\displaystyle y4=f\left(4\right)\text{and}y5=f\left(5\right)}$ and then transmit the sequence y1, y2, y3, y4, y5. If the received sequence is 1,4, 0, 6, 0 and there is at most one error, determine what the correct sequence is. (Again, remember that you are working mod 7- that will simplify the arithmetic slightly.)

If ${\displaystyle B=p+5\text{and}A=p^2+10p+1}$, find an expression that equals ${\displaystyle 2B+3A}$ in standard form.

What is the remainder when the polynomial ${\displaystyle x+x^3+x^9+x^{27}+x^{81}+x^{243}}$ is divided by ${\displaystyle x^2-1}$?

How can I prove that ${\displaystyle 4^n+15n-1\equiv 0\left(\text{mod}9\right)}$?

Is the number ${\displaystyle n^2+n+41}$ a ' for all positive integers n?

If ${\displaystyle x-y=3}$, then what is ${\displaystyle x^3-y^3-9xy}$?

Is it true that if a, b, c are in arithmetic progression, and if the polynomials ${\displaystyle (b-c)x^2+(c-a)x+(a-b)}$ and ${\displaystyle 2(c+a)x^2+(b+c)x}$ have a common root, then ${\displaystyle a^2,c^2,b^2}$ must be un arithmetic progression also?

In how many ways can 8 people be seated in a row if a.) there are no restrictions on the seating arrangement(answer is ${\displaystyle 8!=40,320}$) , b.) persons A and B must sit next to each other? (Answer: 10,080) , c.) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other? (Answer: 1,152) , d.) there are 5 men and they must sit next to one another? (Answer: 2,880) , e.) there are 4 married couples and each couple must sit together? (Answer: 384). I need to see the steps to solve questions 10b to 10e. I quickly looked through the corresponding chapter on combinatorial analysis and I went over topics such as the multinomial theorem, the basic principle of counting, the binomial theorem, and representing the number of possible combinations of n objects taken r at a time as "n choose r" .

Factorial, but with addition [duplicate]
Is there a notation for addition form of factorial?
${\displaystyle 5\ne 5\times 4\times 3\times 2\times 1}$
Thats

How many times a particular number is a zero for a given polynomial. For example, in the polynomial function ${\displaystyle f\left(x\right)={(x-3)}^4(x-5){(x-8)}^2}$, the zero 3 has multiplicity 4, 5 has multiplicity 1, and 8 has multiplicity 2. Although this polynomial has only three zeros, we say that it has seven zeros counting multiplicity.

To find: The sum of the given arithmetic sequence.
The sequence ${\displaystyle \sum n=190(3-2n)}$

${\displaystyle (0.1,1.45),(0.2,2.20),(0.4,3.76)}$ and ${\displaystyle (0.5,432)}$ datas are given. P(0) is approximated using Neville's Method as an approximation of (0). So compute the sum ${\displaystyle Q_{3.7}+Q_{2.2}+P\left(0\right)}$. (Use 5 digs rounding arithmetic)
A.4.2569
B.4.4232
C.4.1509
D.4.0534

${\displaystyle 1+5+9+\cdots +85}$
Write the arithmetic series in summation notation.

Find the sum of the arithetic series.
${\displaystyle \sum _{i=1}^{10}2i}$

Find the indicated term in each arithmetic sequence
${\displaystyle a_9;a_1=-5,d=4}$

Find the sum of each arithmetic series
${\displaystyle \sum _{n=1}^{13}(3n+5)}$

Find ${\displaystyle S_{20}}$ for the arithmetic series ${\displaystyle 5+10+15+\dots ..}$