Question

@wio I have a project on inventing my own polynomial identity and corresponding numerical proof. Can you help? I'm really lost on this one.

Answer 1

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.
You must:
• Label and display your new polynomial identity
• Prove that it is true through an algebraic proof, identifying each step
• Demonstrate that your polynomial identity works on numerical relationships
WARNING! No identities used in the lesson may be submitted. Create your own. See what happens when different binomials or trinomials are combined. Below is a list of some sample factors you may use to help develop your own identity.
• (x – y)
• (x + y)
• (y + x)
• (y – x)
• (x + a)
• (y + b)
• (x2 + 2xy + y2)
• (x2 – 2xy + y2)
• (ax + b)
• (cy + d)

Answer 2

I don't expect you to do this for me, I just keep ending up with a polynomial thing that doesn't work when you plug in numbers. Beyond which, the project asks you to identify what it's good for, and I have noooo idea.

Answer 3

Want me to go ahead and pick a few?

Answer 4

Btw, we studied these in the lesson, so I can't use them:
Difference of Squares
Difference of Cubes
Square of a Binomial
Sum of Cubes
Pythagorean Theorem

Answer 5

How about I use this one and square it? (x^2 – 2xy + y^2)

Answer 6

Square of a trinomial?

Answer 7

@wio You still there? :)

Answer 8

You can do that, but it will be messy.

Answer 9

And trinomials would be like \(x+y+a\)

Answer 10

Okay, which would you suggest?

Answer 11

what you are doing is essentially squaring a square binomial.

Answer 12

(cy + d)(y - x), maybe?

Answer 13

You can do that. I think you can do whatever one you want.

Answer 14

I tried another one and it wasn't numerically provable...but I'll give this one a whirl. How does this look?
cy^2 + yd - cxy - dx

Answer 15

that is correct.

Answer 16

Okay, what values do you suggest I assign? 1, 2, 3 and 4?

Answer 17

Sure.

Answer 18

just pick some numbers and plug them in.

Answer 19

x = 4, y = 3, c = 2, d = 1.

Answer 20

(2)(3^2) + (3)(1) - (2)(4)(3) - (1)(4)

Answer 21

21 - 24 - 4 = -7?

Answer 22

Um...where does that even get me? Lol.

Answer 23

I'm sorry to be high maintenance, I just don't know what the next step is...

Answer 24

now plug them into the factored form and see if you get the same number.

Answer 25

Oh, I see. Thanks! Doing so now.

Answer 26

OMG it works. Yay!!

Answer 27

Okay, so last question - what would this be good for, mathematically? Just - a shortcut?

Answer 28

@wio I think I'm done :) Thanks!

Answer 29

Any ideas on a name? :)

Answer 30

Nope

Answer 31

Foil No More, lol.

Answer 32

There is no particularly good name for this one.