I need a lesson over simplifying polynomials | Mathematics | Page 1

loud totem Oct 22, 2023, 9:33 PM

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Because my math teachers sucked and when we did algebra they just skipped over to equations, average public school smh
Been like that since 7th grade yeah my teachers suck

loud dewBOT Oct 22, 2023, 9:33 PM

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worn notch Oct 22, 2023, 9:33 PM

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It doesnt matter if your teacher sucks

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Everything is online

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Dont be discouraged

loud totem Oct 22, 2023, 9:34 PM

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Ik, js
I can't be here doing calculus 1 stuff and struggling at simplifying derivatives
Not like I'm discouraged

worn notch Oct 22, 2023, 9:35 PM

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Polynomials are what happens when we multiply together an unknown with itself (a few times) and add/subtract numbers

loud totem Oct 22, 2023, 9:35 PM

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Uh huh

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For example, 2x*x+6?

worn notch Oct 22, 2023, 9:39 PM

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Yes

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This is a polynomial

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Or

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$x \times x \times x + 2 \times x + 3$

fair pebbleBOT Oct 22, 2023, 9:40 PM

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Daddy_314

worn notch Oct 22, 2023, 9:40 PM

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Which we can simply denote by

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$x^3+2x+3$

loud totem Oct 22, 2023, 9:40 PM

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Can I do it myself

fair pebbleBOT Oct 22, 2023, 9:40 PM

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Daddy_314

loud totem Oct 22, 2023, 9:40 PM

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Aw ok

worn notch Oct 22, 2023, 9:40 PM

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To simplify a bit

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As you can notice

worn notch Oct 22, 2023, 9:40 PM

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loud totem For example, 2x*x+6?

What you wrote here

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Is equal to $2x^2+6$

fair pebbleBOT Oct 22, 2023, 9:41 PM

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Daddy_314

worn notch Oct 22, 2023, 9:41 PM

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The maximum power of x is 2 in your polynomial

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But in mine it's 3

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This is called the degree of the polynomial

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Which is the maximum number of times the unknown was multiplied by itself

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So your polynomial is of degree 2

loud totem Oct 22, 2023, 9:42 PM

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Ok

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So that's why ax²+bx+c is called a 2nd degree polynomial and equaling it to 0 makes it an equation?

worn notch Oct 22, 2023, 9:44 PM

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Exactly

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Sometimes we call them quadratics

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But thats like a cat called Tom because we know it well

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The neighbors can just call it cat

loud totem Oct 22, 2023, 9:45 PM

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If we have x²*x² does that simplify to x⁴?

worn notch Oct 22, 2023, 9:45 PM

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Yes

loud totem Oct 22, 2023, 9:46 PM

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So it's just adding up the exponents

worn notch Oct 22, 2023, 9:46 PM

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It works just like numbers

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Literally

loud totem Oct 22, 2023, 9:46 PM

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Right

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,tex If $\sqrt{x}=x^{\frac{1}{2}}$, Is $\sqrt{x}*x=x^{\frac{3}{2}}$?

fair pebbleBOT Oct 22, 2023, 9:49 PM

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Bamtryfoil (noisificated)

worn notch Oct 22, 2023, 9:49 PM

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Well

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What you wrote is true

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But anytime you see powers that are not integers

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The object is not a polynomial

loud totem Oct 22, 2023, 9:50 PM

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Taking notes

worn notch Oct 22, 2023, 9:50 PM

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So $f(x) = \sqrt{x} x$ is not a polynomial in x

fair pebbleBOT Oct 22, 2023, 9:50 PM

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Daddy_314

loud totem Oct 22, 2023, 9:50 PM

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Right

worn notch Oct 22, 2023, 9:51 PM

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But if I take $u = \sqrt{x}$ for example
Then $f(x) = x \times \sqrt{x} = u^2 \times u = u^3$ is a polynomial in u

fair pebbleBOT Oct 22, 2023, 9:51 PM

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Daddy_314

worn notch Oct 22, 2023, 9:51 PM

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So the rules are clear, powers of the variable have to be integers

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Now the reason you may be traumatized by polynomials specifically

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Is because one thing we try to do to them is factor them

loud totem Oct 22, 2023, 9:52 PM

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That's the thing

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And i have to do that for taking limits because some take us to false indeterminations

worn notch Oct 22, 2023, 9:52 PM

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Yes

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This typically happens when both polynomials

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In the numerator and denominator

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Share a common factor

loud totem Oct 22, 2023, 9:53 PM

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And I know the l'hopital rule but I need to know factorization for 1st degree bachelor

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So I can do the limits

worn notch Oct 22, 2023, 9:54 PM

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Yeah dont use that

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For polynomials

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This is a strong tool

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No need for that

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Everytime

loud totem Oct 22, 2023, 9:55 PM

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L'hopitals just makes very long derivatives when we have to do product and division rules

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Factorization just should work better

worn notch Oct 22, 2023, 9:55 PM

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Depends

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If its only polynomials then

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Either of them works

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But if you struggle

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Then you need to train

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On factorization properly

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Which means you need to do exercises and struggle

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Until you make some progress

loud totem Oct 22, 2023, 9:57 PM

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Yeah it's gonna be hard

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Can you teach me a little

worn notch Oct 22, 2023, 9:59 PM

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Yes

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What does factorization mean ?

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Can you factor the number $20$ ?

fair pebbleBOT Oct 22, 2023, 10:00 PM

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Daddy_314

loud totem Oct 22, 2023, 10:02 PM

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I think no

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Is it (10*10)

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10+10*

worn notch Oct 22, 2023, 10:02 PM

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loud totem Is it (10*10)

Yes

worn notch Oct 22, 2023, 10:02 PM

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loud totem 10+10*

No

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Factor means to transform something into a multiplication

loud totem Oct 22, 2023, 10:03 PM

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Ohh

worn notch Oct 22, 2023, 10:03 PM

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Preferably of simpler terms

loud totem Oct 22, 2023, 10:03 PM

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Isn't 10*10 100 tho

worn notch Oct 22, 2023, 10:03 PM

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I could always write $20 = 20 \times 1$ but this would be useless

fair pebbleBOT Oct 22, 2023, 10:03 PM

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Daddy_314

worn notch Oct 22, 2023, 10:03 PM

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Well yes but

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20 = 2 * 10

loud totem Oct 22, 2023, 10:04 PM

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I see

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Is there a way to factor any integer?

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Without having to do it just on the spot?

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Because then what about factoring big numbers like 89076

worn notch Oct 22, 2023, 10:06 PM

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Yes !

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Good question

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This is a big deal in maths

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It's called

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The fundamental theorem of arithmetic

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Any number can be written as a product of smaller, elementary numbers

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And these elementary numbers cannot be decomposed into products

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They're called prime numbers

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And the reason we like to factor polynomials, is that it allows us to find its roots

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A root of a polynomial is a number that makes it equal to zero

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For example of we take
$P(x) = x^2-9$

If x = 3,
P(3) = 9-9 = 0

fair pebbleBOT Oct 22, 2023, 10:09 PM

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Daddy_314

worn notch Oct 22, 2023, 10:09 PM

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Therefore 3 is a root

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To know this, We could factor P into a product of smaller easier polynomials

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We could prove that
$p(x) = (x+3)(x-3)$

fair pebbleBOT Oct 22, 2023, 10:09 PM

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Daddy_314

loud totem Oct 22, 2023, 10:11 PM

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I'm lost from this point

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I understand roots but not that

worn notch Oct 22, 2023, 10:19 PM

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Its normal

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I didnt give you details

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Its just the concept for now

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The main idea

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This needs to be done later on

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I have to go for now

loud totem Oct 22, 2023, 10:57 PM

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Alright

loud totem Oct 23, 2023, 1:51 PM

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I'm out of school

worn notch Oct 23, 2023, 2:01 PM

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Welcome

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Ok here's your task:

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Recite the multiplication tables of 0, 2 and 6

loud totem Oct 23, 2023, 3:48 PM

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Mult tables?
Like multiples?

worn notch Oct 23, 2023, 6:38 PM

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Multiplication tables

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0 * 0 = 0
0 * 1 = 0

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Etc

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You will see why

loud totem Oct 23, 2023, 11:52 PM

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0 = 0

2= 2,4,6,8,10,12,14,16,18,20...

6= 6,12,18,24,30,36,42,48,54,60...

slender fable Oct 24, 2023, 12:53 AM

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@loud totem hey! if you ask your question i could probably help as well! its a bit difficult for me to read through all of that lol

worn notch Oct 24, 2023, 6:39 AM

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loud totem 0 = 0 2= 2,4,6,8,10,12,14,16,18,20... 6= 6,12,18,24,30,36,42,48,54,60...

Please list all the 0 operations

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0 * 0 = 0
0 * 1 = 0
0 * 2 = 0

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Until 10

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The follow-up question to that one:

What do you notice

loud totem Oct 24, 2023, 2:55 PM

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worn notch The follow-up question to that one: What do you notice

Multiplication is reiterated sum?

loud totem Oct 24, 2023, 3:13 PM

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I actually don't know

worn notch Oct 24, 2023, 4:49 PM

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loud totem Multiplication is reiterated sum?

Yes

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Is it possible to have two numbers (I call them A and B)

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Such that

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$A \times B = 0$

fair pebbleBOT Oct 24, 2023, 4:50 PM

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Daddy_314

worn notch Oct 24, 2023, 4:50 PM

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And neither A nor B is zero ?

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Same question with A , and B if
$A \times B = 12$
Can you deduce the values of A and B ?

loud totem Oct 24, 2023, 4:51 PM

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I say no

fair pebbleBOT Oct 24, 2023, 4:51 PM

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Daddy_314

loud totem Oct 24, 2023, 4:52 PM

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There are multiple values for B and a but I can say that A could be six and B 2
Or 3 And 4 for example

worn notch Oct 24, 2023, 4:52 PM

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Or A = 0.5 and B = ...

worn notch Oct 24, 2023, 4:52 PM

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worn notch Is it possible to have two numbers (I call them A and B)

What about this?

loud totem Oct 24, 2023, 4:53 PM

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worn notch Is it possible to have two numbers (I call them A and B)

No

loud totem Oct 24, 2023, 4:54 PM

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worn notch Or A = 0.5 and B = ...

B = 24

worn notch Oct 24, 2023, 6:15 PM

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So now that we saw this

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This tells you how special 0 is

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If a product of two numbers is 0, one of these numbers at least must be zero

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This is not the case for any other number (like 12)

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This fact is the basis on which everything around polynomials is built

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Why ?

If for example $x \times (x+2) = 0$
Then we can be dead sure that $x = 0$ or that $x+2=0$

fair pebbleBOT Oct 24, 2023, 6:17 PM

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Daddy_314

loud totem Oct 24, 2023, 8:45 PM

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Yup

loud totem Oct 24, 2023, 9:18 PM

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I got that

loud totem Oct 25, 2023, 2:47 PM

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slender fable <@887074205500334090> hey! if you ask your question i could probably help as wel...

Ok cool

loud totem Oct 25, 2023, 8:26 PM

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Just learned ruffini's method of factoring 3rd degree + polynomials

worn notch Oct 25, 2023, 10:02 PM

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Ruffini ? Sounds like a cat name

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Can you explain this method

oak pulsar Oct 25, 2023, 11:28 PM

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idk man just get better

#I need a lesson over simplifying polynomials