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    Polynomial Identities

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    Polynomial Identities and Proofs
    Essential Questions:
    • How can polynomial identities be proven?
    • What can polynomial identities apply to beyond just polynomials?

    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.
    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 or even word document.
    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
    Create your own using the columns below. See what happens when different binomials or trinomials are combined. Square one factor from column A and add it to one factor from column B to develop your own identity.

    Column A Column B
    (x − y) (x2 + 2xy + y2)
    (x + y) (x2 − 2xy + y2)
    (y + x) (ax + b)
    (y − x) (cy + d)

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    Solution Preview

    • How can polynomial identities be proven?

    Polynomial identities can be proven by performing operations such as FOIL (First Outside Inside Last), multiplying using the Box method, or by substituting numbers into the variables.

    When proving an identity, the goal is to transform one side of the identity so that it is identical to the other side of the identity. The work shown should not transform both sides of the identity simultaneously. Instead, we start with one side of the identity (such as the left side) and use known operations such as addition, subtraction, multiplication, division, or FOIL to prove that the side we started with (left side) is equal to the other side (right side) of the identity.

    Note that an ...

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