Algebra Precalculus – Convert between Cartesian and polar equations and their graphs

I am asked to convert $ (x + 2) ^ 2 + y ^ 2 = 4 $ in a polar equation, and then confirm on my calculator. I converted it to $ r = 4cos ( theta) $but the graphics do not look the same at all. If someone could explain how to check for each answer and why the graphics do not look the same, that would be great.

Algebra Precalculus – A polynomial in disguise. What am I missing conceptually?

The equation family:

$$ e ^ { frac {s} { ln (x)}} = e ^ { frac {t} { ln (1-x)}} $$ $ s, t in Bbb N $

$ x in Bbb A (0,1) $

Is a family of polynomial equations in disguise. It can be rewritten as:

$ x ^ t = (1-x) ^ s. $

Draw the parameter space for all $ s, t $ is equivalent to drawing the null set of this family of polynomial equations in $ (0,1) ^ 2. $

On the one hand, to draw points that satisfy $ x ^ t = (1-x) ^ s, $ yields points of the form $ P ( Bbb A, Bbb A). $

On the other hand, points are plotted that satisfy $ e ^ { frac {s} { ln (x)}} = e ^ { frac {t} { ln (1-x)}}, $ yields points of the form $ P ( Bbb A, y) $

The $ y- $Coordinates can be numbers in $ Bbb Q, Bbb T, Bbb A. $

If these two representations are equivalent to the family of algebraic equations, then why can not I determine the nature of the algebraic equations? $ y- $Coordinate? What do I miss here? Are they in fact fundamentally different equations, but between them is a map in the form of algebraic manipulations?

Algebra Precalculus – Should the set of irrational numbers be subdivided into more subsets?

The real numbers are divided into two subcategories. the rational numbers, those that can be written as a relationship, and the irrational ones, those that can not.

But we know that many of the irrational numbers, though they can not be written as proportion or fraction, can still be written as continue Fraction. So my question is, are there any more subdivisions of irrational numbers into those that can be written as a regular continuous fraction (whose nominators and denominators follow a regular pattern instead of being random), and those that are or should be tilted? By "present" I mean the amount that has been assigned a specific letter and that is officially recognized as a set of numbers.

Algebra precalculus – Define the domain of $ f (x) = sqrt {x ^ 2-4} $

I'm supposed to be the area of $ f (x) = sqrt {x ^ 2-4} $,

I arrived $ (- 2, infty) $ while the textbook solution is $ (- infty infty) $,

To get to my solution, I set the radicand to greater than or equal to zero:

$ x ^ 2 + 4 ge0 $

$ x + 2 ge0 $ # Square root of each page

$ x ge-2 $

This will give me the domain as $ (- 2, infty) $,

Why is the domain actually $ (- infty infty) $?

Algebra precalculus – solving two equations in three variables

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Algebra Pre-calculus – However, the attempt to solve this inequality has not progressed at all.

The problem is, that $ x, y, z $ are right breaks, and each of them is greater than zero.

Given: $ x + y + z = 2 $

Must prove $ frac {x. y. z} {(1-x). (1-y). (1-z)} 8q,

I tried to solve this with AM $ geq $ GM inequality.

Attempt :

$ frac { frac {1-x} {x} + frac {1-y} {y} + frac {1-z} {z}} {3} $ $ geq $ $ biggl ( frac {(1-x). (1-y). (1-z)} {x.y.z} biggl) ^ {1/3} $

What should I do to calculate the value of? $ frac {1} {x} + frac {1} {y} + frac {1} {z} $ ?

Algebra precalculus – $ x_1 x_2 x_3 x_4 + x_2 x_3 x_4 x_5 + …… + x_n x_1 x_2 x_3 = 0 $ What is $ n $?

Can someone please help me to understand the following problem?Enter image description here

It's really hard for me to understand this problem. What I understood is that I have to find a natural number $ n $ for which the equation holds, whatever the values ​​of $ x_i $ That's impossible. Because of this, some conditions seem to be on $ x_i $ s are necessary ..

Algebra Pre-calculus – Problems in interpreting information to develop mathematical functions

Information about interpreting

Context: assignment of the university code

So far I've developed two models based on this information:

For the surface zone I have $ T (l) = frac {-11} {45} l + 24 $ and for the deep zone I have $ T (l) = 2 $ from where $ T $ is the summer temperature in degrees Celsius and $ l $ is the latitude in degrees (where $ 0 $ is the equator and $ 90 are the poles).

I have trouble interpreting this information to develop a linear model for the summer temperature of seawater ($ T $) in the thermocline zone. I know that this model will be different from the last two, because it depends on depth (not latitude), but the lack of data really makes me nervous.

Any guidance would be very grateful.

Algebra Pre-Calculus – How to multiply both sides of $ frac {5} {X_1-X_2}> 10 $ with $ X_1-X_2 $ if $ X_i $ are independent of each other as random variables?

Suppose we have random variables $ X_1 $ and $ X_2 $ they are independent and distributed identically.

Suppose I am interested in the inequality $ frac {5} {X_1-X_2}> 10 $,

How can I multiply both sides of this inequality? $ X_1-X_2 $

Especially since $ X_1 $ and $ X_2 $ are random variables, I do not know if $ X_1 -X_2 $ is positive or negative. So I do not know if I have to reverse the sign of inequality or not.

Furthermore, a statement would please $$ frac {5} {X_1-X_2}> 10 $$ THEN AND ONLY THEN, IF $$. 5> X_1-X_2 $$ be right?

I think it would be just an attempt to prove it. Suppose that $ .5> X_1-X_2 $, Then it has to be that $ X_1> X_2 $ and we can rearrange to get $ frac {5} {X_1-X_2}> 10 $ by $ X_1-X_2 $ , (So the IF direction applies)

And for the only if directionagain if $ frac {5} {X_1-X_2}> 10 $ by $ X_1-X_2 $ then $ X_1-X_2 $ must be positive, since a negative number can not be greater than $ 10and we get the result by rearranging.

Algebra Precalculus – Number of possible integer values ​​of x for which a given expression is an integer

How many integers, $ x $Check the following

begin {equation *}
frac {x ^ 3 + 2x ^ 2 + 9} {x ^ 2 + 4x + 5}
end {equation *}

is an integer?

I did it:

begin {equation *}
frac {x ^ 3 + 2x ^ 2 + 9} {x ^ 2 + 4x + 5} = x-2 + frac {3x + 19} {x ^ 2 + 4x + 5}
end {equation *}

but I can not continue.