Polynomial Long Division

Polynomial Long Division

Polynomial long division is the most common method used to find the quotient and remainder when dividing two polynomials. This method is similar to the long division we perform with integers.

Preparing for Long Division

Before starting the division, there are a few things to prepare:

1. Order Terms:

   Write the dividend ($P(x)$) and the divisor ($Q(x)$) in descending order of variable powers (from highest power to lowest).

2. Complete Terms:

   If any term with a specific power is missing (its coefficient is zero), still write that term with a coefficient of 0 as a placeholder. This is crucial for keeping the columns aligned during subtraction.

   Example:

   If $P(x) = 4x^4 + 17x^3 - 3x + 1$, the $x^2$ term is missing.

   So we write it as $P(x) = 4x^4 + 17x^3 + 0x^2 - 3x + 1$.

3. Set Up Division:

   Write the division in long division format, with $P(x)$ (the completed form) inside the division symbol and $Q(x)$ outside.

Steps for Long Division

The division process is performed step-by-step as follows:

1. Divide: Divide the first term of $P(x)$ by the first term of $Q(x)$. Write the result as the first term of the quotient ($H(x)$) above the line.
2. Multiply: Multiply the quotient term just obtained by the entire divisor $Q(x)$.
3. Subtract: Write the result of the multiplication below $P(x)$, aligning like terms, then subtract it from $P(x)$ to get a temporary remainder.
4. Bring Down: Bring down the next term from $P(x)$ next to the temporary remainder to form a new polynomial.
5. Repeat: Repeat steps 1-4 with this new polynomial until the degree of the temporary remainder is less than the degree of the divisor $Q(x)$.

Long Division Example

Divide $P(x) = 4x^4 + 17x^3 - 3x + 1$ by $Q(x) = x^2 + 4x - 1$.

1. Preparation:

   • $P(x) = 4x^4 + 17x^3 + 0x^2 - 3x + 1$ (complete the $x^2$ term)
   • $Q(x) = x^2 + 4x - 1$

2. Division Process:

   $$\begin{array}{l} \qquad\qquad\qquad 4x^2 + x \\ x^2+4x-1\overline{\big)4x^4 + 17x^3 + 0x^2 - 3x + 1} \\ \qquad\qquad\underline{-(4x^4 + 16x^3 - 4x^2)} \\ \qquad\qquad\qquad\qquad x^3 + 4x^2 - 3x \\ \qquad\qquad\qquad\quad\underline{-(x^3 + 4x^2 - x)} \\ \qquad\qquad\qquad\qquad\qquad\qquad -2x + 1 \\ \end{array}$$

3. Step-by-Step Explanation:

   Iteration 1:

   • Divide: Divide the first term $4x^4$ by the first term of the divisor $x^2$:

     $$\frac{4x^4}{x^2} = 4x^2$$

     Write $4x^2$ as the first term of the quotient.

   • Multiply: Multiply $4x^2$ by the divisor $x^2 + 4x - 1$:

     $$4x^2(x^2 + 4x - 1) = 4x^4 + 16x^3 - 4x^2$$

   • Subtract: Subtract the result from the initial polynomial:

     $$(4x^4 + 17x^3 + 0x^2) - (4x^4 + 16x^3 - 4x^2) = x^3 + 4x^2$$

   • Bring Down: Bring down the next term ($-3x$) to get the new polynomial:

     $x^3 + 4x^2 - 3x$

   Iteration 2:

   • Divide: Divide the first term of the new polynomial $x^3$ by the first term of the divisor $x^2$:

     $$\frac{x^3}{x^2} = x$$

     Write $+x$ as the next term of the quotient.

   • Multiply: Multiply $x$ by the divisor $x^2 + 4x - 1$:

     $$x(x^2 + 4x - 1) = x^3 + 4x^2 - x$$

   • Subtract: Subtract the result from the current polynomial:

     $$(x^3 + 4x^2 - 3x) - (x^3 + 4x^2 - x) = -2x$$

   • Bring Down: Bring down the next term ($+1$) to get the temporary remainder:

     $$-2x + 1$$

   Stop: The degree of the remainder ($-2x+1$, degree 1) is less than the degree of the divisor ($x^2+4x-1$, degree 2), so the division stops.

4. Result:

   • The Quotient ($H(x)$) is $4x^2 + x$.
   • The Remainder ($S(x)$) is $-2x + 1$.

5. Writing in Division Algorithm Form:

   Based on the division algorithm, we can write the result as:

   • Fraction Form:

     $$\frac{4x^4 + 17x^3 - 3x + 1}{x^2 + 4x - 1} = 4x^2 + x + \frac{-2x + 1}{x^2 + 4x - 1}$$

   • Multiplication Form:

     $$4x^4 + 17x^3 - 3x + 1 = (x^2 + 4x - 1)(4x^2 + x) + (-2x + 1)$$

This long division method might seem lengthy, but it is very systematic and reliable for all types of polynomial division.

Exercise

Find the quotient polynomial $H(x)$ and the remainder polynomial $S(x)$ after dividing $P(x) = x^3 - x + 9$ by $Q(x) = x^2 - 2x + 3$.

State the result in the form $P(x) = Q(x) \cdot H(x) + S(x)$.

Answer Key

Complete $P(x)$ to become $x^3 + 0x^2 - x + 9$.

• Quotient: $H(x) = x + 2$

• Remainder: $S(x) = 3$

• Division Algorithm Form:

  $$x^3 - x + 9 = (x^2 - 2x + 3)(x + 2) + 3$$