P(x)=D(x)Q(x)+R(x) D is a nonzero polynomial. The remainder is zero or has degree less than D. P(x)D(x)=Q(x)+ R(x)D(x) D(x) 0 Arrange terms in descending powers, inserting zero coefficients for missing powers. Divide the leading terms, multiply, subtract and repeat. Example: divide 3x3-4x2+5x+4 by x2-3x **Quotient term** | **Subtract** | **Remainder** 3x | 3x(x2-3x) | 5x2+5x+4 5 | 5(x2-3x) | 20x+4 Quotient: 3x+5. Remainder: 20x+4. Check: 3x3-4x2+5x+4=(x2-3x)(3x+5)+(20x+4). The remainder theorem finds only the remainder; use division when the quotient is also needed.

\(D\) is a nonzero polynomial. The remainder is zero or has degree less than \(D\).

\(\dfrac{\mathrm{P}(x)}{D(x)}=\mathrm{Q}(x)+\dfrac{R(x)}{D(x)}\qquad D(x)\ne0\)

Arrange terms in descending powers, inserting zero coefficients for missing powers. Divide the leading terms, multiply, subtract and repeat.

Example: divide \(3x^3-4x^2+5x+4\) by \(x^2-3x\)

Quotient: \(3x+5\). Remainder: \(20x+4\).

Check: \(3x^3-4x^2+5x+4=(x^2-3x)(3x+5)+(20x+4)\).

The remainder theorem finds only the remainder; use division when the quotient is also needed.

Long Division of Polynomials

Long Division of Polynomials