[PPT] The Polynomial Remainder Theorem

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The Polynomial Remainder Theorem for Algebra II students.

Presentation Transcript

The [Polynomial] Remainder Theorem With its Proof:

The [Polynomial] Remainder Theorem With its Proof Teacher Lecture Algebra II

The Polynomial Remainder Theorem:

The Polynomial Remainder Theorem The Polynomial Remainder Theorem states: The remainder of the division of a polynomial f(x) by a linear polynomial x – a is equal to f(a) . In particular, x – a is a divisor of f(x) if and only if f(a) = 0.

Video: Introduction the Polynomial Remainder Theorem:

Video: Introduction the Polynomial Remainder Theorem

Example:

Example The objective here is to understand the algebraic procedural flow of the application of the Remainder Theorem. We will go over this step-by-step.

Practice Problems:

Practice Problems What is the remainder R when the polynomial p(x) is divided by ( x + 1)? P(x) = 5 x 4 + 6 x 3 + x 2 +2 x + 2 1.1 Is ( x + 1) a factor of p(x) ? Does ( k – 2) divide into k 3 – k 2 – k – 2 evenly?

Question #1:

Question #1 What if The Remainder Theorem was never found? Would you find it practical to use synthetic division? How do you think mathematicians perceive The Remainder Theorem versus unpractical methods?

Proof of The Remainder Theorem:

Proof of The Remainder Theorem We will be able to explain every step within this proof!

Question:

Question A proficient proof reader is able to prove every line. If I were to give you another theorem of ease, then would you be able to justify/argue every step of the proof? *I would review with you the proof before you would be tested on it.

Another use of The Polynomial Remainder Theorem:

Another use of The Polynomial Remainder Theorem

Question:

Question We just saw another application of The Remainder Theorem. We are able to construct a polynomial with a missing coefficient given that we have a linear factor of the polynomial. Within a simple idea like this, are you able to think of any applications in the world that parallel with this idea (e.g., machinery)?

Summary:

Summary The [Polynomial] Remainder Theorem Proof of the Remainder Theorem Applications of the Remainder Theorem