Expert Answers to Algebra 2 Problems

Find an equation of the plane tangent to the following surface at the given point.
${\displaystyle yz{e}^{xz}+14=0;(0,-7,2)}$

Is there a number that is exactly 1 more than its cube?

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.)

Use a linear approximation (or differentials) to estimate the given number.
${\displaystyle {\left(1.999\right)}^5}$

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
${\displaystyle a_n=\frac{1}{2n+3}}$

Rearrange this expression into quadratic form, ${\displaystyle a{x}^2+bx+c=0}$, and identify the values of a, b, and c.
${\displaystyle =0.20=\frac{x^2}{75}-x}$

Line AB contains points A(4,5) and B(9,7). What is the slope of ${\displaystyle \overline{AB}}$?

Find all polar coordinates of point ${\displaystyle P=(6,{31}^{\circ })}$.

Find two positive real numbers whose product is a maximum. The sum is 110.

Use Formula to find the curvature.
${\displaystyle y=x^3}$

Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a.
${\displaystyle f\left(x\right)=\frac{1}{x+5}}$
${\displaystyle a=5}$

To rationalize the denominator of the expression ${\displaystyle \frac{1}{\sqrt{2}+x}}$

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

${\displaystyle \text{Condense the following expression into a single logarithm using the properties of logarithms. Assume x>0 and }x\ne 6}$
${\displaystyle {\log }_5(x^2-8x+12)-2{\log }_5(x-6)}$

Write the expression as a sum, difference, or product of logarithms. Assume that all variables represent positive real numbers:
${\displaystyle {\log }_b\left(\frac{m^5{p}^9}{n^3{b}^7}\right)}$

Review
1) ${\displaystyle 5^{x-7}=3^{2x+1}}$
2) ${\displaystyle 11e^{-4x}=21}$

Solve for xx.
${\displaystyle {\log }_2\left(x\right)+{\log }_2(x-12)=6}$