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Polynomial Operations

The Hook

The Math Train.

What is it?

Expressions with many terms (poly=many, nomial=name/term) like x³ - 4x² + x - 5.

Time Travel (History)

Persian mathematician Al-Khwarizmi laid the groundwork for solving these equations, giving us the word 'Algebra'.

Cheat Code

Synthetic Division is a shortcut for dividing by (x-k). It's magic.

Practice This Now!

The Field Guide

In Plain English: A polynomial is like a train where every car is a power of x. You can add trains (combine like terms), multiply them (FOIL on steroids), or divide them. The 'Degree' is just the biggest exponent—it tells you how curvy the graph is.

In The Real World: Roller Coaster Design. Engineers use cubic polynomials to design smooth tracks that don't snap your neck.

How To Do It

  1. Standard Form: Arrange terms from highest power to lowest.
  2. Add/Sub: Only combine terms with the exact same exponent.
  3. Multiply: Distribute every term in the first group to every term in the second group.
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Booby Trap!

The Freshman Dream Error: Thinking (x+3)² is x²+9. NO! You must write (x+3)(x+3) and foil to get x²+6x+9.

Real World Challenge
"A box has width x, length x+2, and height x-1. Write a polynomial for the volume."
The Logic: V = x(x+2)(x-1). Foil (x+2)(x-1) -> x²+x-2. Multiply by x -> x³+x²-2x.

Training Drills

Synthetic Div
1 | 1 -3 2 | 1 -2 ----------- 1 -2 0
Coefficients only.