MEDALS GIVEN! 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. You must:
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
im lost.. what is the first step to doing this?
They have given a bunch of samples. Each one is a polynomial. They want you to combine one or more polynomials and come up with an equation. Here I have chosen to create a polynomial by multiplying (x+a) and (x+a) (Note: (x+a) is one of the samples listed.) (x+a)(x+a) = x^2 + ax + ax + a^2 = x^2 + 2ax + a^2 The left hand side is (x+a)^2. Therefore the polynomial identity is: (x+a)^2 = x^2 + 2ax + b^2 They are asking for a whole lot of other things, but this is the gist of the math part.
hmm i think im starting to understand
so the polynomial would be (x+a)^2 = x^2 + 2ax + b^2 ?
The right side is the polynomial. But the equation is an identity involving a polynomial. An identity means you can put various values for the variables on the left and right and no matter what you choose the left side should match the right side. To prove it numerically, I can choose x = 3 and a = 2 and calculate the left side and right side separately and prove they are identical (hence an identity)
Sorry about the b. That should be a. (x+a)^2 = x^2 + 2ax + a^2 Let x = 3 and a = 2 LHS = (3+2)^2 = 5^2 = 25 RHS = 3^2 + (2)(2)(3) + 2^2 = 9 + 12 + 4 = 25 LHS = RHS and therefore we have proven the polynomial identity.
oh i see so the polynomial itself is (x+a)^2 and the identity is x^2 + 2ax + a^2 ?
Either side is a polynomial: (x+a)^2 is a polynomial -- a second degree polynomial. x^2 + 2ax + a^2 is also a second degree polynomial We are saying they are one and the same. They are identical. So the identity is: (x+a)^2 = x^2 + 2ax + a^2 Identities are equations where we are saying that the left side equals the right side no matter what value you choose for the variables and constants in the identity. And we just proved it with a numerical example. We also derived the identity algebraically by actually multiplying (x+a) and (x+a) and arriving at x^2 + 2ax + a^2 And (x+a) times (x+a) is (x+a)^2
ok so my identity is (x+a)^2 = x^2 + 2ax + a^2
and any number subbed for x would make them equal the same?
Yes any value for x and any value for the constant a will make the LHS = RHS. Only then we can call it an identity. We cannot call x + 4 = 2x an identity. It is just an equation. It holds true ONLY when x = 4. It is not true for any other value for x. But an identity is true for all values.
I see i see. so is that all they're asking for? an identity?
They are asking for a whole lot of stuff but this is the gist of the math part.
ok. i have to eat now, ill be back in a second. please help when i come back!
I don't know how much more help I can offer for this particular question because the math part is done. PS: In the question they say: WARNING! No identities used in the lesson may be submitted. If the identity (x+a)^2 = x^2 + 2ax + a^2 or something equivalent is already covered in the lesson then you have to come up with some other identity.
ok i checked and (x+a)^2 is not in the lesson so were good there
it says make the world make you want to buy it? what does that mean?
For the past few days the reply notifications, messages, tagging etc. are not working properly and so unless I am on this window I won't know if you have posted any replies or tagged me. They are asking for a whole lot of non-mathematical stuff. They want you to "sell" the product like you were selling a car or something. In other words they want you to tell all the benefits of having this identity and how much it will make life easier, etc. These is not math anymore and so I won't be of much help for the rest of the questions.
oh ok then thanks! by the way you've helped me before on another account, but i forgot my pass so i made a new one
you're probably the most helpful on this website, the mods never help me lol
you are welcome.
wait im still confused on one thing. what are the steps to proving this identity? can you list them?
ok never mind I've got it
help me feather
whats the answer?!
@kenndallnicole did you firgure this out!?
divide the standard quadratic form by a that is the answer