Division of Polynomials

Quotient and Remainder in Integer 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 $4$:

• As a mixed number, $\frac{7}{4}$ = $1 \frac{3}{4}$. The integer quotient is $1$, and the remainder is $3$.
• For the division algorithm, we write the equivalent statement $7 = 4 \cdot 1 + 3$.

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:

The equivalent fractional form is:

The polynomial identity $P=QH+S$ is valid for every $x$. The fractional form is defined only at values where $Q(x)\neq0$. In both forms, the degree of $S(x)$ is less than the degree of $Q(x)$, or $S(x)=0$ when there is no 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 $S(x) = 2$

The division result can be written in two forms according to the algorithm:

1. Fractional form:

   $$\begin{aligned} &\frac{x^3 + 4x^2 + 5x + 8}{x + 3} \\ &= x^2 + x + 2 + \frac{2}{x + 3} \end{aligned}$$

2. Multiplication form:

   $$\begin{aligned} &x^3 + 4x^2 + 5x + 8 \\ &= (x + 3)(x^2 + x + 2) + 2 \end{aligned}$$

The remainder $S(x)=2$ has degree $0$, which is less than the degree of the divisor $x+3$, namely $1$. This is the condition that makes the division complete.

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

Expand the right side:

Distribute $x+3$:

Distribute $x$ and $3$:

Group like terms:

The result is:

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

Exercise

Divide $P(x)=2x^3-3x^2+4x-5$ by $Q(x)=x-2$. State the quotient and remainder, then verify the result with the division algorithm. Reconstruct the dividend as $(x-2) \cdot H(x) + S(x)$ and confirm that every term matches.

Worked Solution

Because the divisor is $x-2$, use $c=2$ in the Horner scheme:

The quotient is $H(x)=2x^2+x+6$ and the remainder is $S(x)=7$. Verify by reconstructing the dividend:

The calculation gives: