Division between polynomials - quotient and remainder

Just as for numbers there is division with remainder, polynomials too can be divided.

Definition — Division between polynomials

Given two polynomials $A(x)$ (dividend) and $B(x)$ (divisor) with $\deg B \le \deg A$, there exist unique $Q(x)$ (quotient) and $R(x)$ (remainder) such that: $A(x) = B(x)\cdot Q(x) + R(x), \qquad \deg R < \deg B.$

Example

$\frac{x^3-2x^2+x-1}{x-3} = x^2+x+4 + \frac{11}{x-3}$ Check: $(x^2+x+4)(x-3) = x^3-3x^2+x^2-3x+4x-12 = x^3-2x^2+x-12$, and $-12+11=-1$. ✓

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Topics: Algebraic calculation
Concepts: Division between polynomials
Skills: Calculating