Master Polynomial Concepts with Comprehensive Resources

Prove the Arithmetic-Geometric Mean Inequality If ${\displaystyle a_1{a}_2,\dots ,a_n}$ are nonnegative numbers, then their arithmetic mean is ${\displaystyle \frac{a_1+a_2+\dots +a_n}{n},}$ and their geometric . The avthmetic-geometic mean is ${\displaystyle \sqrt{n}\{a_1,a_2,\dots a_n\}.}$ The arithmetic-geometric mean equality states that the geometric mean is always less than or equal to the arithmetic mean. In this problem we prove this in the case of two numbers andy. (a) If x and y are nonnegative and ${\displaystyle x\le y,then{x}^2\le y^2.}$ [Hint: First use Rule 3 of Inequalities to show that ${\displaystyle x^2\le xy\text{and}xy\le y^2.}$ ] (b) Prove the arithmetic-geometric mean inequality ${\displaystyle \sqrt{xy}\le \frac{x+y}{2}}$

1. Explain with numerical examples what Real Numbers and Algebraic Expressions are. 2. Explain with numerical examples Factoring and finding LCMs (least common multiples). Explain factoring of a larger number. 3. Explain with numerical examples arithmetical operations (addition, subtraction, multiplication, division) with fractions 4, Explain with numerical examples arithmetical operations (addition, subtraction, multiplication, division) with percentages 5. Explain with numerical examples exponential notation 6. Explain with numerical examples order (precedence) of arithmetic operations 7. Explain with numerical examples the concept and how to find perimeter, area, volume, and circumference (use related formulas)

For each sequence, decide whether it could be arithmetic,
$a)\hspace{0.17em}\hspace{0.17em}25,\hspace{0.17em}\hspace{0.17em}5,\hspace{0.17em}\hspace{0.17em}1,\hspace{0.17em}\hspace{0.17em}...$
$b)\hspace{0.17em}\hspace{0.17em}25,\hspace{0.17em}\hspace{0.17em}19,\hspace{0.17em}\hspace{0.17em}13,\hspace{0.17em}\hspace{0.17em}...$
$c)\hspace{0.17em}\hspace{0.17em}4,\hspace{0.17em}\hspace{0.17em}9,\hspace{0.17em}\hspace{0.17em}16,\hspace{0.17em}\hspace{0.17em}...$
$d)\hspace{0.17em}\hspace{0.17em}50,\hspace{0.17em}\hspace{0.17em}60,\hspace{0.17em}\hspace{0.17em}70,\hspace{0.17em}\hspace{0.17em}...$
$e)\hspace{0.17em}\hspace{0.17em}\frac{1}{2},\hspace{0.17em}\hspace{0.17em}3,\hspace{0.17em}\hspace{0.17em}18,\hspace{0.17em}\hspace{0.17em}...$

The explanation of how to multiply polynomialswhen neither is monomial with an examplw:
${\displaystyle (2x^2+4y^3)(3x^3+4y^2)}$

The missing terms for the given polynomal p, q, r, and s such that
$q\ne 0,r\ne 0ands\ne 0,\frac{p}{q}-:\frac{r}{s}=\frac{ps}{qr}$

We need find:
The real zeros of polynomial function form an arithmetic sequence
$f(x)=x^4-4x^3-4x^2+16x.$

Determine whether the sequence is arithmetic. If so, identify the common difference 1, 2, 4, 8, 16, …

Suppose that $\{a1,a2,a3,\cdots \}$ is an arithmetic sequence with common difference d. Explain why $\{a1,a3,a5,\cdots \}$ is also an arithmetic sequence.

Enzo's profit over the last few months has been failing. In December, he made $900. In January, his profit was $500, and in March, he lost $300. is the sequence that represents his profit arithmetic? if it is, what is the recursive formula for the sequence?

Consider the following sequence.
$s_n=2n-1$
(a) Find the first three terms of the sequence whose nth term is given.
$s_1=$
$s_2=$
$s_3=$
(b) Classify the sequence as arithmetic, geometric, both, or neither. arithmetic, geometric bothneither
If arithmetic, give d, if geometric, give r, if both, give d followed by r. (If both, enter your answers as a comma-separated list. If neither, enter NONE.)

Write the first six terms of the arithmetic sequence ${\displaystyle a_n=a_{n-1}+4,a_1=-7}$

Find the sum of the first n terms of the arithmetic sequence.$-9+(-12)+(-15)+...$ (to 10 terms) Find the sum of the first 100 terms of an arithmetic sequence with 15th term of 86 and first term of 2.

FInd the zeroes of the polynomials
${\displaystyle p\left(x\right)=x^2+5x+6}$

In each of the​ following, list three terms that continue the arithmetic or geometric sequences. Identify the sequences as arithmetic or geometric. a. 2, 6, 18, 54, 162 b. 1, 8 ,15, 22, 29 c. 11, 15, 19, 23, 27

True or false: If the sequence $a,b,c\text{ is arithmetic, then the sequence }2a,2b,2c$ is geometric. Justify by giving example.

Calculate: Polymomial equation with real coefficients taht has roots $5,4-3iis{x}^3-13x^2+65x-125=0$

Given the following function: ${\displaystyle f\left(x\right)=1.01e^{4x}-4.62e^{3x}-3.11e^{2x}+12.2e^x-1.99}$ a)Use three-digit rounding frithmetic, the assumption that ${\displaystyle e^{1.53}=4.62}$, and the fact that ${\displaystyle e^{nx}={\left(e^x\right)}^n}$ to evaluate ${\displaystyle f\left(1.53\right)}$ b)Redo the same calculation by first rewriting the equation using the polynomial factoring technique c)Calculate the percentage relative errors in both part a) and b) to the true result ${\displaystyle f\left(1.53\right)=-7.60787}$

The arithmetic mean (average) of two numbers c and d is given by $\bar{x}=\frac{c+d}{2}$
The value $\bar{x}$ is equidistant between c and d, so the sequence c,$\bar{x}$, d is an arithmetic sequence. inserting k equally spaced values between c and d, yields the arithmetic sequence $c,{\bar{X}}_1,{\bar{X}}_2,{\bar{X}}_3,{\bar{X}}_4,...,{\bar{X}}_k,d$. Use this information for Exercise.
Insert four arithmetic means between 19 and 64.