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1. Remainder Theorem

Instead of doing long polynomial division, the Remainder Theorem provides a shortcut to calculate the remaining value directly when a polynomial f(x) is divided by a linear divisor.

If f(x) is divided by (x - a), then Remainder (R) = f(a)
If f(x) is divided by (ax - b), then Remainder (R) = f(b/a)
If f(x) is divided by (x + a), then Remainder (R) = f(-a)

2. Factor Theorem

The Factor Theorem links polynomial roots directly to division remainders. It acts as the core baseline for factoring cubic equations.

Theorem Condition:
A linear term (x - a) is a true factor of the polynomial f(x) if and only if the absolute remainder equates to zero:
f(a) = 0

3. Synthetic Division Framework

Synthetic division is a shorthand method of polynomial division where you drop variables and calculate using only the numerical coefficients.

Table Layout Design for f(x) = Ax³ + Bx² + Cx + D divided by (x - a):

a A B C D
--- a × A a × (New Value) a × (New Value)
A B + aA Quotient Coefficients Remainder

Rule: Drop down the first coefficient, multiply by the root 'a', add along columns, and repeat. The final boxed value is your remainder.

4. Rational Root Theorem

If a polynomial equation has fractional or rational roots, this theorem defines exactly where they must come from based on the leading coefficient and the constant term.

For Polynomial: P(x) = a_n x^n + ... + a_0
Possible Rational Roots = ± (Factors of Constant Term a_0) / (Factors of Leading Coefficient a_n)

How to use it to solve cubic equations (f(x) = 0):

  1. List all integer factors of the constant number at the end. Call these factors p.
  2. List all integer factors of the very first coefficient. Call these factors q.
  3. Write down all possible fractions form of p / q.
  4. Test these specific values inside f(x) using the factor theorem until you find one that gives 0. This stops you from guessing blind random numbers!