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Polynomial Operations

Overview

The Math Train.

What is it?

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

History

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

Key Idea

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

Practice This Topic

Concept Guide

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.

Real-world example: 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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Common Pitfall

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.

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

Practice Examples

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