# State the fundamental theorem of arithmetic.

Question
Polynomial arithmetic
State the fundamental theorem of arithmetic.

2021-02-17

Every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
For example, $$20=2^2 \times 5 = 2\times2\times5$$

### Relevant Questions

Prove the Fundamental Theorem of Arithmetic. Every integer than 1 is a prime or a product of primes. This product is unique, exept for the order in which the factors appear.

Consider the "clock arithmetic" group $$(Z_{15}, \oplus)$$ a) Using Lagrange`s Theotem, state all possible orders for subgroups of this group. b) List all of the subgroups of $$(Z_{15}, \oplus)$$

Use Part 2 of the fundamental Theorem of Calculus to find the derivatives.
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{1}}}^{{{x}}}}{\ln{{t}}}{\left.{d}{t}\right.}$$
Use Part 2 of the fundamental Theorem of Calculus to find the derivatives.
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{0}}}^{{{x}}}}{e}^{{\sqrt{{{t}}}}}{\left.{d}{t}\right.}$$
Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$\displaystyle{P}{\left({x}\right)}={3}{x}^{{4}}-{5}{x}^{{3}}+{x}^{{2}}-{3}{x}+{5}{N}{S}{K}{Q}{\left({x}\right)}={\left({\left({\left({3}{x}-{5}\right)}{x}+{1}\right)}{x}-{3}\right)}{x}+{5}$$ Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial $$\displaystyle{R}{\left({x}\right)}={x}^{{5}}—{2}{x}^{{4}}+{3}{x}^{{3}}—{2}{x}^{{3}}+{3}{x}+{4}$$ 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 ofa polynomial using synthetic division?
Consider the following sequence. $$\displaystyle{s}_{{n}}={2}{n}−{1}$$ (a) Find the first three terms of the sequence whose nth term is given. $$\displaystyle{s}_{{1}}={N}{S}{K}{s}_{{2}}={N}{S}{K}{s}_{{3}}=$$ (b) Classify the sequence as arithmetic, geometric, both, or neither. arithmeticgeometric 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.)
Prove the Arithmetic-Geometric Mean Inequality If $$\displaystyle{a}_{{1}}{a}_{{2}},\ldots,{a}_{{n}}$$ are nonnegative numbers, then their arithmetic mean is $$\displaystyle{\frac{{{a}_{{1}}+{a}_{{2}}+\ldots+{a}_{{n}}}}{{{n}}}},$$ and their geometric . The avthmetic-geometic mean is $$\displaystyle\sqrt{{{n}}}{\left\lbrace{a}_{{1}},{a}_{{2}},\ldots{a}_{{n}}\right\rbrace}.$$ 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 $$\displaystyle{x}\leq{y},{t}{h}{e}{n}{x}^{{2}}\leq{y}^{{2}}.$$ [Hint: First use Rule 3 of Inequalities to show that $$\displaystyle{x}^{{2}}\leq{x}{y}{\quad\text{and}\quad}{x}{y}\leq{y}^{{2}}.$$ ] (b) Prove the arithmetic-geometric mean inequality $$\displaystyle\sqrt{{{x}{y}}}\leq{\frac{{{x}+{y}}}{{{2}}}}$$
$$\displaystyle\sum\ {n}={1}\ {100}\ {\left({6}\ -\ {12}\ {n}\right)}$$
$$\displaystyle\sum\ {n}={1}\ {80}\ {2}{n}\ -\ {5}={6080}$$