Division of Polynomials

Basic Concept of Polynomial Division

Polynomial division is similar to the division of integers we are familiar with. When we divide one number by another, we get a quotient and a remainder.

For example, when dividing 7 by 4:

• $\frac{7}{4}$ can be written as $1 \frac{3}{4}$ (1 remainder 3)
• This can also be written as $7 = 4 \cdot 1 + 3$

Here:

• 7 is the dividend
• 4 is the divisor
• 1 is the quotient
• 3 is the remainder

The same concept applies to polynomials.

Polynomial Division Algorithm

The division algorithm states the relationship between the dividend polynomial, the divisor, the quotient, and the remainder.

If $P(x)$ (the dividend) and $Q(x)$ (the divisor) are two polynomials, with $Q(x) \neq 0$, then there exist unique polynomials $H(x)$ (the quotient) and $S(x)$ (the remainder) such that:

or it can be written as:

with the condition that the degree of $S(x)$ is less than the degree of $Q(x)$, or $S(x) = 0$ (zero remainder).

Terminology:

• $P(x)$: Dividend Polynomial
• $Q(x)$: Divisor Polynomial
• $H(x)$: Quotient Polynomial
• $S(x)$: Remainder Polynomial

Illustrative Example:

The division of $x^3 + 4x^2 + 5x + 8$ by $x + 3$ yields:

• Quotient $H(x) = x^2 + x + 2$
• Remainder $S(x) = 2$

This can be written in two forms according to the algorithm:

1. Fractional form:

   $$\frac{x^3 + 4x^2 + 5x + 8}{x + 3} = x^2 + x + 2 + \frac{2}{x + 3}$$

2. Multiplication form:

   $$x^3 + 4x^2 + 5x + 8 = (x + 3)(x^2 + x + 2) + 2$$

Note that the degree of the remainder ($S(x)=2$, degree 0) is less than the degree of the divisor ($x+3$, degree 1).

Verifying the Division Algorithm

We can prove the correctness of the second form above by multiplying the quotient by the divisor and then adding the remainder.

Prove that $x^3 + 4x^2 + 5x + 8 = (x + 3)(x^2 + x + 2) + 2$.

Let's expand the right side:

Since the right side equals the left side, the equation is proven true.