Chapter 2: Polynomials

1. Polynomials and Roots

Definition: A polynomial P(x) of degree n is defined as:

P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

Concept of Roots:

A value c is a root (or zero) of the polynomial if P(c) = 0.

Root Condition: If P(c) = 0, then (x - c) is a factor of P(x).

2. Synthetic Division

Synthetic division is a fast, shorthand method of dividing polynomials when the divisor is a linear factor in the form (x - c). Instead of writing out all the variables, we only use the coefficients.

Worked Example

Divide P(x) = 2x³ - 5x² + 3x - 7 by (x - 2)

Identify 'c': Since we are dividing by (x - 2), our c = 2.
List the coefficients: The coefficients of P(x) are 2, -5, 3, and -7. (Note: If a term is missing, you must put a 0 in its place!)

2	2	-5	3	-7
	↓	(2×2) 4	(2×-1) -2	(2×1) 2
	2	-1	1	-5

How to read the result:

The last number (-5) is the Remainder.
The other numbers (2, -1, 1) are the coefficients of your new quotient polynomial. The new polynomial will always be one degree lower than the original.
Final Answer: Quotient = 2x² - 1x + 1, Remainder = -5

3. Essential Theorems

Remainder Theorem: When a polynomial P(x) is divided by (x - c), the remainder is simply the value of P(c).

Factor Theorem: (x - c) is a factor of P(x) if and only if P(c) = 0.