Expert Answers to Algebra 2 Problems

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?

To calculate: Polynomial equation with real coefficients that has roots ${\displaystyle 3i,3-i}$

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?

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 ${\displaystyle x^3-y^3-9xy}$?

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

When 10 times the square root of 48 is written in simplest radical form, the result is k times the square root of 3. What is the value of k?

Guys need your help For the function, determine if it is a polynomial, rational function, both, or neither
${\displaystyle f\left(x\right)={(6x^5-4)}^2}$
a. Polynomial function only
b. Rational function only
c. Both polynomial and rational function

Find a rational function ${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ with range ${\displaystyle f\left(\mathbb{R}\right)=[0,1]}$ (Thus ${\displaystyle f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}}$ for all ${\displaystyle x\in \mathbb{R}}$ for suitable polynomials P and Q, where Q has no real root).
I have no clue where to begin.

Let ${\displaystyle R(t_1,\dots ,t_n)ϵ\mathbb{R}(t_1,\dots ,t_n)}$ be a rational function in n variables. Suppose ${\displaystyle R_i=R(1,1,\dots ,1,t_i,1,1,\dots ,1)}$ is a polynomial in ${\displaystyle t_i}$ for ${\displaystyle 1\le i\le n}$. Is it true that ${\displaystyle R(t_1,\dots ,t_n)}$ is a polynomial?

How does one analytically determine the integer values of a rational function $f(x)=\frac{40-8x}{8x+2}mod1$ where x is an element of the rationals?

I was trying to convert a rational function ${\displaystyle \mathbb{R}\left(X\right)}$ to a polynomial of type ${\displaystyle \mathbb{R}|X,X^{-1}|}$ but I failed. I searched internet and ${\displaystyle \mathbb{R}|X,X^{-1}|}$ has its own name!: "Laurent polynomials".
a. Every Laurent polynomial is a combination of rational functions. Is every Laurent polynomial possible to be equal to a single rational function?
b. Are there methods to convert rational functions to Laurent polynomials? How are they?

I have to create a rational function with the characteristics: 3 real zeros(1 of them of multiplicity 2). y-intercept 1. vertical asymptote at x=-2 and x=3. oblique asymptote y=2x+1. Ive

R(z) be a rational function satisfies|R(z)|=1 when |z|=1. Prove that R(z) can be written as ${\displaystyle R\left(z\right)=c{z}^m\prod _{n=1}^k\frac{z-a_n}{1-\overline{a_n}z}}$, c is a constant and ${\displaystyle 0<\left|a_n\right|\ne 1}$. Hint: try to prove that if ${\displaystyle z_0}$ is a zero of R(z) then ${\displaystyle \frac{1}{\overline{z_0}}}$ is a pole of R(z)

A rational function is a function f of the form ${\displaystyle \frac{p}{q}}$ where p and q are polynomial functions. The domain of f is ${\displaystyle \{x\in \mathbb{R}:q\left(x\right)\ne 0\}}$. Prove every rational function is continuous.
I have previously proved that every polynomial function ${\displaystyle p\left(x\right)=a_0+a_{1}x+\dots +\dots a_n{x}^n}$ is continuous on ${\displaystyle \mathbb{R}}$. Could I use a theorem that states ${\displaystyle \frac{f}{g}}$ is continuous at ${\displaystyle x_0}$ if ${\displaystyle g\left(x_0\right)\ne 0}$? Then ${\displaystyle p\left(x\right)}$ (which is already proved to be continuous) over ${\displaystyle q\left(x\right)}$ would be continuous?