Question

You are going to design an advertisement for a new polynomial identity that you are going to invent. Your goal for this activity is to demonstrate the proof of your polynomial identity through an algebraic proof and a numerical proof in an engaging way! Make it so the whole world wants to purchase your polynomial identity and can't imagine living without it!

You may do this by making a flier, a newspaper or magazine advertisement, making an infomercial video or audio recording, or designing a visual presentation for investors through a flowchart or PowerPoint.

Answer 1

First thing you need is a polynomial identity.

Answer 2

Do you have one in mind?

Answer 3

No I dont

Answer 4

do you know what an identity is?

Answer 5

..no

Answer 6

an identity is an equation that is always true for all allowed values in the domain

Answer 7

something like 1 = 1, 10 = 10, x = x are all identities

Answer 8

So I need to create one?

Answer 9

yes

Answer 10

another identity: x+x = 2x

another one: 23x - 8x = 15x

Answer 11

all of the equations posted are true for all values of x

Answer 12

How about 13x+4x= 17x?

Answer 13

that's an identity, good

Answer 14

whatever you plug in for x, it will be a true equation

Answer 15

Alright what do I do next?

Answer 16

something more complicated

x^2 + 3x - 2x + 9 = x^2 + x + 9

Answer 17

well the algebraic proof for 13x+4x= 17x is simply showing that both sides are equal to the same thing

Answer 18

numerically, you just plug in various numbers to show that the equation is always true

Answer 19

So now I would create a more complex one?

Answer 20

you can create one as simple or as complex as you want

Answer 21

all they care about is that you have an identity of some sort (that simplifies to a polynomial)

Answer 22

So is the one I did fine ? Or would it be better if I did a more complex one?

Answer 23

it might be better to go with a complicated one since you have more work to show

Answer 24

Then you have to design some sort of flier or presentation to show the world this identity and you have to make it like a sales pitch it seems

Answer 25

I'm REALLY bad at math so could you help me make it step by step?

Answer 26

which identity are you going with?

Answer 27

(ax+b)

Answer 28

remember, an identity is an equation that's always true (for all of the variables' values)

Answer 29

oh so I'll use the one I made earlier

Answer 30

here's a good somewhat complicated identity

Answer 31

(x+3)^2 = x^2 + 6x + 9

Answer 32

no matter what you plug in for x, you'll get the same number on both sides (when you evaluate), so you'll get a true equation for all real numbers x

Answer 33

Alright so we got that, now what?

Answer 34

you have to prove that (x+3)^2 = x^2 + 6x + 9 is true

Answer 35

both algebraically and numerically

Answer 36

So pick a number to be x?

Answer 37

for the numeric proof, yes

Answer 38

4

Answer 39

ok replace every x with 4 and see if you get a true equation

Answer 40

4+3)^2 = 4^2 + 6(4) + 9

Answer 41

(4+3)^2 = 4^2 + 6(4) + 9

good

Answer 42

then evaluate each side

Answer 43

what does ^ stand for ?

Answer 44

^ means exponent

Answer 45

so 8^2 means 8 exponent 2 or 8 squared

Answer 46

\(\Large 8\wedge 2 = 8^2\)

Answer 47

ohhh okay

Answer 48

It's mainly notation for typing in text only (like in a text document or on a forum like this) and calculators also use this notation too

Answer 49

49 is the first half?

Answer 50

left side, yes

Answer 51

I got 41 for the other side

Answer 52

oh no i did it wrong hold on

Answer 53

I got 49 got the other side

Answer 54

you should get 49 for both sides, so be extra sure you get 49 for the RHS (right hand side)

Answer 55

I did :)

Answer 56

What now?

Answer 57

use other numbers to show it works for them as well

I'd say 3 examples is probably good enough but you can go as high as you want

Answer 58

this will all be for the numeric form of the proof

Answer 59

Alright I'll do that now.

Answer 60

7+3)^2 = 7^2 + 6(7) + 9

Answer 61

100 on the first side

Answer 62

I got 100 for both sides

Answer 63

(7+3)^2 not 7+3)^2

Answer 64

good, 100 = 100 is the final result, so that verifies the identity for x = 7

Answer 65

whoops

Answer 66

Here is one way to algebraically prove it's an identity

Answer 67

(x+3)^2 = x^2 + 6x + 9

(x+3)(x+3) = x^2 + 6x + 9

x(x+3)+3(x+3) = x^2 + 6x + 9

x^2+3x+3x+9 = x^2 + 6x + 9

x^2 + 6x + 9 = x^2 + 6x + 9

Answer 68

so that takes care of the algebraic approach

Answer 69

Alright, thank you !

Answer 70

you're welcome

Answer 71

Thats everything right ?

Answer 72

you just have to make the presentation, but that's it in terms of the math portion really

Answer 73

Alright thank you ! Will you be on here tomorrow?

Answer 74

I should be, so feel free to message whenever

Answer 75

I will, again thank you sooo much !

Answer 76

sure thing