**Assignment Instructions:**

**Ready?**

Need Even More Proof?

Are you ready to dive into this new polynomial identity and try it out for yourself? Do not be shy, have fun!

If you are not certain that this is true, I will provide you with even more proof using numerical proof!

x = 2, a = 3, b = 1

(ax + b)(x + a) = ax^2 + a^2x + bx + ab

(3*2 + 1)(2 + 3) = 3(2^2) + 2(3^2) + (1*2) + (3*1)

(6 +1)(5) = 3(4) + 2(9) + 2 + 3

7*5 = 12 + 18 + 2 + 3

35 = 35

This numerical proof is correct! As you can see, this identity is correct, whether you use algebraic proof or numerical proof.

By: Briahna Vlcek

**4.08 Polynomial Identities and Proofs**

You can prove this identity using an algebraic method called the distributive property, otherwise known as foil.

First, you need to foil (ax + b)(x + a)

You use foil on the first term (ax) and you end up with ax^2 + a^2x.

Then, you use foil on the second term (b) and end up with bx + ab

Add those together and you get, ax^2 + a^2x + bx + ab.

Need Proof?

Below, you will see the newest polynomial identity:

(ax + b)(x + a) = ax^2 + a^2x + bx + ab

Identity Exposed

Are you ready to be surprised? This advertisement shows you the newest way to identify polynomials. This will expand your knowledge and make you feel smarter as quick as saying "123!" Hold onto your hats, folks, because you are going to be blown away!

New Identity Starts Here

It's time to show off your creativity and marketing skills!

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:

• 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)