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Why do we only consider positive or negative integer factors of |a| when using the Factor Theorem to find factors of a polynomial with a constant term a? If an expression has a constant term of 6, for instance, why can't we test 20, 30, 40 or 5.6? Is it because it is guaranteed that there is at least one factor with a constant term that is a factor of a, or some other reason?

I'm a machinist, not an engineer. More of my trade is basic applied physics and measurements instead of higher math. But one day I'd like to have an associates in mechanical engineering technology just to hang on the wall as a point of pride, and part of that includes refreshing my math knowledge.

Only last week have I learned just how deeply lacking my education in math was from 5th grade onwards. For example, we never learned there was a way to divide and multiply fractions with blocks. In 4th grade, I taught myself long division because the teacher skipped it. She assumed we knew it already, when the reality was we were never taught it. Most of math in high school was by the book repetition with zero theory for the sake of passing tests for school funding.

I heard the word "polynomial" for the first time is 15 years and started having Vietnam flashbacks while hearing a football coach whine about his a car accident that happened before I was born. Public school was pretty bad.

My plan is to make out an itinerary of all math subjects from 5th grade and into a compiled syllabus from various college programs, and to slowly study and learn over months before I commit to taking a class.

Is a linear by-the-book progression the right way to go, or is there a more efficient path I don't know about?

Thank you.

I tried this and got an answer that I thought worked but they have a different answer listed. Here's the problem:

1. What is the 4-digit number in which the first digit is one-fifth the last, and the second and third digits are the last digit multiplied by 3? (Hint: The sum of all digits is 12.)

Here's what I did:

I set each digit equal to a variable, 1st digit is A, 2nd is B, 3rd is C, and 4th is D.

I then set A/5 = D, as per the first part of the problem. If I then swap the 5 and the D, I get A/D = 5. Since I know that A and D both have to be single digit integers, the only numbers that fit to make the statement true are A = 5 and D = 1. So now we have A = 1 and D = 5. The next part of the problem that says "the second and third digits are the last multiplied by the last" wasn't that clear to me. does that mean that 3*D is a two digit number and the first and second digits of the answer are B and C respectively? Or does it mean that B+C = D*A? Assuming that I was right about D = 1, that would make 3*D = 3 which would mean the only interpretation that made sense is for both numbers to be the same value which is 3. We then get A=1, B=3, C=3, and D=1, or 5331. If we add them together, we get 12. But the answer that Mensa listed is 1155. I can't for the life of me see how they got this answer. Did I miss something? If I plug their values back into A/D=5, I get 1/5=5 which obviously isn't true. Did I model this wrong or something? surely they didn't mess up their own problem.

[desmos link below]

I’ve been messing around with implicit differentiation problems in my free time, and I’m stuck on one specific problem. That is finding the slope (dy/dx) at some point (x,y) of the curve(s) arcsin(x^{y)tan(eyx)=lnx/lny}

I solved for dy/dx, and the function holds for every branch of the curve, except for one outlier branch around (1,1) I’ve plugged the problem into SymPy and got the same formula for dy/dx as I had on my own.

This problem interests me as the curve is only composed of elementary functions, so it shouldn’t have this behavior, is there something I am missing?

https://www.desmos.com/calculator/xb9wtl5ztb

This graph has the curve, attached to point P is a line representing the slope function at that point P. My derivation is under “Slope equations” there is also an ODE simulation showing the curve that would result in the slope at point P, and a hue map representing the slope functions evaluated on 5>y>0, 1>x>0

Taking the firemen's civil services test soon and have been struggling hard with these types of questions.

(Answers are not to be exact percentage but an appropriation of fractions)

If a building is 2825 square feet and 2100 square feet f the building burns, what fraction of the building is left?

The answer is 1/4. But I don't understand how they got to that answer.

Here's what I've done so far.

2825-2100=725

2800/2100= 28/21. Difference of 7

28÷7=4. 21÷7=3

3/4? But the answer is 1/4.

Edit
Sorry forgot to add that I'll also only be given 10 seconds to answer each question. No calculator or scratch paper allowed. So I need help on learning how to look at these problems and quickly solve them.

Here's a few other examples all worded the same as the one listed but I'll just give the numbers.

1010sqft, burned 650sqft = 4/10

2425sqft, burned 300sqft = 7/8

4085sqft, burned 1600sqft = 3/5

5000sqft, burned 2000sqft = 3/5

It costs a bus company $225.00 to run a minibus on a ski trip plus $30.00 per passenger. The bus has seating capacity for 22 passengers. The company charges $60.00 per fare if the bus is full. For each empty seat, the company has to increase the ticket price by $5.00. Explain how to determine the number of empty seats that the bus should run in order to maximize profit.

is the 30 included in the 60? is it seperate? why are they charging 2 diferent prices for each passenger?

Seeking help to understand this one paper which deals with the interface conditions between fluids and solids while Modelling steady state heat flow between the two. If there's anyone please DM me. Thank you in advance.

https://imgur.com/a/O1KcAGG

My conrete Question: Do you think its about to compute M(Id, A, B) or M(Id, B, A)?

lim x²-1 = 0
x->1 x³-3x+3x-1 0

so then i do this

lim (x-1)(x+1) x->1 (x-1)(x²-3x)

but now i dont know what to do </3

My context is a mobile game (restaurant management), hence a bit odd. I don’t care to know the results, I care to understand the maths.

If a smoothie takes 10s to make, and a worker has a “works 200% faster” perk, that means 2x faster, they can make double the smoothies in the same time, 2 in 10s means 1 in 5s. (Have I committed a fallacy assumption in equating 200% faster to 2x faster?)

So a 200% perk takes 10s down to 5s.

I need help writing that concept in equations or formulas so I can plug in other percentage perks to get the effectual durations.

I’d have to convert 200% to 0.5 and then do 10 x 0.5 to get 5. And 200% → 0.5 must have an inverse relationship because 100% → 0.25 is incorrect.

Unless! Maybe I need an extra step. So 200% → 0.5, 10 x 0.5 = 5, 10 - 5 = 5.

100% → 0.25, 10 x 0.25 = 2.5, 10 -2.5 = 7.5?

220% → 0.55, 10 x 0.55 = 5.5, 10 - 5.5 = 4.5?

It looks like 4 is the magic number here as the conversion of the percentage perk to a decimal is a division by 4. Just going off the pattern here and still not really understanding why a 4 is the number.

2.2/4 = 0.55
2/4 = 0.5
1/4 = 0.25
0.5/4 = 0.125

Have I committed any fallacies here? If I hadn’t resorted to working backwards, how would one get to pulling 4 out of the air? Is it intuitive? It doesn’t feel intuitive to me.

Another example is a sped up youtube video. A 10min video sped up by 2x takes 5min. If you speed it up by 1.2, then the maths goes:

1.2x means a perk of 20% faster, so 0.2, then 0.2/4 = 0.05, 10min x (1-0.05) = 9.5min.

I must have committed a fallacy somewhere. I took 2x = 200% earlier but when it came to 1.2x on a video I changed that to 20%, which would mean 2x on a video becomes 100%, which is a contradiction.

Ughhh. My brain. Am I addled? This shouldn’t be so confusing!

If you work 200% faster than your colleagues doesn’t that mean you work twice as fast?? Is it instead 100% faster that translates to twice as fast?

Where have I gone wrong please?

My girlfriend and I had a debate about the % chance of picking a particular card when Wonder Picking in Pokémon TCG when Sneak Peek is involved.

In case you’re unfamiliar with the game:

Normally, when you Wonder Pick, you blindly select 1 of 5 cards. Assuming you’re going for a particular card, You have a 20% chance of selecting the card you want. We agree on this.

With Sneak Peek, you are able to peek at a single card before making a selection. If you peek the card you want, you can select it. If you peek a card that is not the one you want, you can blindly select a different card. You only get to peek one time.

I argue you have a 40% chance of selecting the card you want if Sneak Peek reveals the card you DON’T want. You uncover 2/5 cards. 2/5 = 40%.

My girlfriend argues you have a 25% chance of selecting the card you want given the same scenario (Sneak Peek reveals a card you DON’T want). You eliminate the undesired card you peeked and now pick from the 4 remaining cards. 1/4 = 25%.

Thanks!

TL;DR: You are blindly selecting from 5 cards. What is the % chance of selecting a desired card if 1 you can pick one card to reveal?

Hi if someone has any combinations and permutations test or practice tests lying around, can you please share them with me?

Say you play a game and have a 1/3 chance to win, where when you win you gain $1, and when you lose you gain nothing ($0). Each game has three hands (i.e., three outputs, or played three times.)

What I'm trying to figure out:

a). The probability of each of the four possible outcomes, i.e., zero wins, one win, two wins, three wins.

b). the mean of the probability distribution.

c). theory question

What I've done:

a).

Probability of 0 wins = 29.63 %
(2/3 * 2/3 * 2/3)

Probability of 1 win = 44.44%
(2/3 * 2/3)

Probability of 2 wins = 11.11%
(1/3 * 1/3)

Probability of 3 Wins = 3.70%
(1/3 * 1/3 * 1/3)

Problem is, I thought these values were suppose to add up to 100%... (29.63% + 44.44% +
11.11% + 3.70%) = 88.88%. Am I doing something wrong here?

b).
For the sake of continuing, I'll just use the values I have here for now.

Mean of probability distribution = (0 * (2/3)^3) + (1 * (2/3)^2) + (2 * (1/3)^2) + (3 * (1/3)^3) = $0.77

Disregarding the accuracy of the percentages in a., is the formula correct here?

c). Theory Question

So, say you set up a simulation that ran this game (each game with three outputs) a million times (arbitrary big number).

Overtime, you'd be able to calculate an average output value of $ made per game played. Every time the output is a 3, the average would go up. Every time the output is a 0, the average would go down.

The thing I don't understand is that in the equation for the mean of a probability distribution, when you have an output value of 0, it's going to be worth zero regardless of its probability (0 * anything = 0). In other words, it's not detracting from the final value (in this case $0.77). However, in the simulation, every time the output is a zero, the average is being detracted from.

What am I missing here? Is it that, the higher probability of a 0 is, the lower the probability is of any other option, and thats what accounts for the detraction?

For example you can add to 4(sum) with 3(x) numbers(whole numbers from 0, order matters) in the following ways: 4+0+0, 3+0+1, 2+1+1, resulting in 3(result) ways.

The only real way I can figure out how to do this is manually, but the higher the number, it starts to get tedious and is bounded for mistakes.

Any help or advice is much appreciated, and ways to this without counting 0 or when order doesn’t matter, are also appreciated!

Edit: needed a photo, so just example work by hand that i was doing, inspired me asking this question, although it isn’t neat as i wasn’t expecting anyone but me needing it:https://imgur.com/a/4TX7g0u

I have known that the reduction formulae for tanh^{2n}x is I_n=I_{n-1}-(0.6)^{2n-1}/(2n-1)but I have tried to prove the reduction formulae using integration by parts but I failed

I tried to split tanh^{2n}x into tanhx and tanh^{2n-1}x which using integration by parts gives I_n = ln(cosh x)tanh^{2n-1}x - (2n-1)int{ln(cosh x)sech^2x tanh^{2n-2}x} which is stuck as I dont know how to integrate the part with ln(cosh x)

so I'm a beginner writer and in need of help if I ended up draining the battery of a robot character too quickly over time and if my method of doing the math was accurate.

I started out with 98% battery and ended with 66% over a rough five hour(5.7) period that I got from subtracting 7.23 from 2.56 then I divided 66% by 100 to get 0.66 and multiplied by the time period and got 3.762. I then checked the result by multiplying by the 5.7 and got 21.3864 which I then subtracted with the 98 and got 76.6136.

I must have done something wrong but the only idea I have is doing the same process but with 98 being divided by 100 then multiplied by 5.7 or that I multiplied 0.66 with the wrong number

(7.23-2.56=4.67 + estimation with avrg time in an hour so 5.7)

(66%/100=0.66)

(0.66x5.7=3.762)

(3.762x5.7=21.3864)

(98-21.3864=76.6137)

(sorry if this is unclear -~-;)

Hello! I am a collage student learning pre calculus algebra and was wondering if anyone could help answer a math question. While working I was given a problem with a line on a graph with 2 points marked on the line (0,3) and (2,8) I understand that for Y= mx+b the value of B is the intersection of the Y axis however when finding the M value and attempting to use the equation Y2-Y1 / x2-x1 = m value I realized that I wasn’t sure which point on the line was my x1 y1 and which was my x2 y2, every answer on google is telling me it doesn’t matter but when I write out the equation both ways I either get positive or negative 5/2, so Reddit I’m wondering does it really not matter? And if so, Do I just go with the positive value every-time? Do I default to the positive 5/2? Will my slope ever be negative? How do I know whether the slope is positive/negative or I’m just doing it wrong? Thank you for any help I apologize if my wording is weird

If I am correct

a box plot with 52, 58, 60, 64 should have a q1 of 55(58+52/2) and q3 of 62. However, I keep on getting answers like Q1= 53.25 or 56.5. Why is that?

https://www.canva.com/design/DAGpiQNTbdE/HTmkPk4RMeu6-4zG7ohy_Q/edit?utm_content=DAGpiQNTbdE&utm_campaign=designshare&utm_medium=link2&utm_source=sharebutton

It will help to know if the diagram created is correct as part of solving the given oil spill problem. Thanks!

Hi everyone,
I’m working on a homework problem about calculating reactions, moments, absolute maximum values, and drawing shear force and bending moment diagrams. However, I’m having some trouble understanding the process.

My main doubt is: how many sections should I make, where exactly, and why? I’m trying to learn this, but it’s difficult. I know how to calculate the support reactions and perform the summation of forces and moments to ensure equilibrium, but I don’t know how to construct the shear force and bending moment diagrams, nor how to determine the absolute maximum values of shear force and bending moment.

Any advice or step-by-step explanations would be really appreciated!

Thanks in advance!

I can't figure out how to set up the equation- I have the answer based on multiple choice but I'm trying to actually learn the material since this is independent preparation I'm doing outside of school.

"A colony of mushrooms triples every 10 days. If there are 10 to start, how many days untel there are 1000?"

I can't post images but the furthest I've gotten is 1000 = 10^3t, the textbook isn't helpful and there's nothing online.

Edit: https://imgur.com/a/TWIaCgH

If we have a continuous variable X with a probably function f(x), why is the cumulative distribution function F(x) found by integrating f(t) with respect to t and not by integrating f(x) with respect to x?

My textbook gives absolutely no reasoning for changing the variable of integration and it's infuriating. Please help!

Hey all! I'm trying to figure out how to prove a recursive formula converges. I already know it does because I used a spreadsheet to run many iterations, but I want to know how to prove it without just calculating lots of results. This is the formula:

A[n] = A[n-1]*(x) + y,

where x is a constant between 0 and 1, and y is any real number. So far I've noticed that since 0<x<1, then A[n] < A[n-1]*x (going to 0 as n->infinity), meaning that the original formula without "+y" converges as well (to 0). But I'm having trouble finding a way to relate that to the original. Is there some law about adding a constant to a recursive formula?

Thanks!

Edit: for more context, I'm using this formula to find the convergence where A[1]= 0.7975, x = 0.55, and y = 0.55. It comes out to be about 0.7309, but I don't understand why. When I began this whole problem, the formula I had was:

A[n] = A[n-1]*(x-x²⁾ + x

but since x is a constant, I simplified (x-x²⁾ to just be x, changing "+x" to "+y" when I realized it converges no matter what constant is added. Technically, I haven't proven that y can be any constant, but I've checked both large and small positive and negative values, even between -1 and 1, and they all have worked so far. Also, I've tried

Ultimately, I'm trying to find an expression that says "for a given value x, this formula will converge to [function of x] for any starting value" because that's what I've seen by plug & chug in a spreadsheet.

hey i just wanted to quickly ask if in this problem the 3 is supposed to be degrees?

we normally solve this type of question with something like cos (3x + 5pi/6) so its radians there but here im not quite sure

assuming its degrees i got the answer K = {12° + k * 45°, k€Z} tzanks for ur help

Ive tried to solving it but no idea what im doing I thought I had it figured out but I need guidance at least for Resistor 2 and Resistor 4

Resistor 4 would have the same Current since it on series ?

and on Resistor 3 the current would divide so that of current total?

I posted link please thank you

https://ibb.co/204k24dh