Horner's Method

Understanding Horner's Method

Horner's method is a more efficient and simpler way to perform polynomial division, especially when the divisor is linear, like $(x-c)$. This method can be considered a simplification of long division because it only involves writing coefficients and more concise multiplication and addition operations.

Comparison with Long Division

To see how Horner's method simplifies the process, let's compare the division of $P(x) = x^3 - 7x + 8$ (or $x^3 + 0x^2 - 7x + 8$) by $x-2$ using both methods:

Long Division:

Horner's Method:

Notice how Horner's method only focuses on the coefficients $(1, 0, -7, 8)$ and the value $c=2$. The bottom row in Horner's method $(1, 2, -3)$ directly gives the coefficients of the quotient $1x^2 + 2x - 3$, and the last number $\boxed{2}$ is the remainder.

This is much more concise than writing out all the variables and powers as in long division.

This method can only be directly used if the divisor is a linear polynomial of degree one, i.e., in the form $(x-c)$ or $(ax-b)$ (which can be transformed).

Preparing to Use Horner's Method

Before performing division with Horner's method, there are several preparation steps:

1. Identify Coefficients of the Dividend ($P(x)$):

   Write down all coefficients of the polynomial to be divided in order, starting from the term with the highest power down to the constant. Ensure no power is skipped. If a term with a certain power is missing, its coefficient is written as 0.

   Example:

   For $P(x) = 2x^3 + 5x^2 + 6$, the coefficient of the $x$ term is 0. So, the coefficients we write in order are: $2, 5, 0, 6$.

2. Identify the Value $c$ from the Divisor ($x-c$):

   Determine the value of $c$ from the divisor polynomial. Remember, if the divisor is $x-c$, the value used is $c$. If the divisor is $x+c$, it is equivalent to $x - (-c)$, so the value used is $-c$.

   Example:

   If the divisor is $x+3$, then $x - (-3)$, so $c = -3$.

3. Set Up the Horner Scheme:

   Draw the Horner scheme or diagram. Place the value $c$ on the left and write the coefficients of $P(x)$ in the top row.

Division Process with Horner's Method

Here are the steps to perform division using the Horner scheme:

1. Bring Down the First Coefficient: Bring down the first coefficient ($a_n$) directly to the result row (the bottom row).
2. Multiply and Place: Multiply the coefficient just brought down by the value $c$. Place the result under the second coefficient ($a_{n-1}$).
3. Add: Add the second coefficient ($a_{n-1}$) to the result from the previous step. Write the sum in the result row, directly below it.
4. Repeat: Repeat step 2 (multiply by $c$) and step 3 (add to the coefficient above it) for all remaining coefficients.
5. Final Result: The last number in the result row is the remainder ($S$). The other numbers in the result row, from left to right, are the coefficients of the quotient polynomial ($H(x)$), starting from the power $n-1$.

Using Horner's Method

Let's divide $P(x) = 2x^3 + 5x^2 + 6$ by $x+3$ using both methods.

Long Division:

Horner's Method:

1. Preparation:

   • Coefficients of $P(x)$: $2, 5, 0$ (for $x^1$), $6$ (constant).
   • The divisor is $x+3$, so $x - (-3)$, thus $c = -3$.

2. Horner Process:

   $$\begin{array}{c|cccc} -3 & 2 & 5 & 0 & 6 \\ & & -6 & 3 & -9 \\ \hline & 2 & -1 & 3 & \boxed{-3} \\ \end{array}$$

3. Scheme Explanation:

   • Bring down the number 2 (coefficient of $x^3$).
   • $2 \times (-3) = -6$. Place -6 under 5.
   • $5 + (-6) = -1$. Write -1 in the result row.
   • $-1 \times (-3) = 3$. Place 3 under 0.
   • $0 + 3 = 3$. Write 3 in the result row.
   • $3 \times (-3) = -9$. Place -9 under 6.
   • $6 + (-9) = -3$. Write -3 (remainder) in the result row on the far right.

4. Result:

   • The last number in the result row is $\boxed{-3}$. This is the Remainder ($S$).
   • The other numbers are 2, -1, 3. These are the coefficients of the Quotient ($H(x)$). Since $P(x)$ is degree 3, $H(x)$ is degree 2.
   • Thus, $H(x) = 2x^2 - 1x + 3 = 2x^2 - x + 3$.

5. Writing in Division Algorithm Form:

   $$P(x) = (x-c) H(x) + S$$

   $$2x^3 + 5x^2 + 6 = (x+3)(2x^2 - x + 3) - 3$$

Exercise

Find the quotient and remainder from the division of $x^4 + 4$ by $x - 1$ using Horner's method and long division.

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

Answer Key

• Dividend: $P(x) = x^4 + 4 = x^4 + 0x^3 + 0x^2 + 0x + 4$.
• Divisor: $Q(x) = x - 1$, so $c = 1$.

Long Division:

Horner's Method:

Result:

• Quotient: $H(x) = x^3 + x^2 + x + 1$.
• Remainder: $S = \boxed{5}$.

Writing in Division Algorithm Form: