Comprehensive Guide to Ruffini's Rule for Polynomial Division

Introduction to Ruffini's Rule

Ruffini's Rule is defined as a highly practical algebraic method used for dividing a polynomial by a binomial specifically of the form $(x - a)$. Beyond division, this method is also extremely useful for finding the roots (zeros) of a polynomial. The process is streamlined by working only with the coefficients of the terms rather than the full algebraic expressions.

Example Problem: Dividing a Third-Degree Polynomial

To demonstrate the application of Ruffini's Rule, consider the following example: we will divide the polynomial $P(x) = 2x^{3} - 5x^{2} + 3x - 7$ by the binomial $(x - 2)$.

Paso 1: Preparing Information and Data

There are two critical sub-steps to prepare for the calculation:

1. Identify the Value of $a$: From the divisor binomio $(x - 2)$, we identify the value of $a$. In this specific form $(x - a)$, if the binomial is $(x - 2)$, then $a = 2$.

2. Organize the Coefficients: List all the coefficients of the polynomial $P(x)$ in order, starting from the term with the highest degree down to the independent term. If the polynomial is missing a term for a specific power of $x$, a $0$ must be placed in its position as a placeholder to maintain alignment. For the polynomial $2x^{3} - 5x^{2} + 3x - 7$, the coefficients are:     * $x^{3}$ coefficient: $2$     * $x^{2}$ coefficient: $-5$     * $x^{1}$ coefficient: $3$     * Independent term: $-7$

These are arranged in a horizontal line for the calculation:

$2 \quad -5 \quad 3 \quad -7$

Paso 2: Commencing the Calculation

The calculation begins by establishing the Ruffini grid. The value of $a = 2$ is placed to the left of a vertical line, and the coefficients are placed to the right.

First, bring down the first coefficient: The leading coefficient ($2$) is moved directly to the bottom line without any modification.

$\begin{array}{c|rrrr} & 2 & -5 & 3 & -7 \\ 2 & \downarrow & & & \\\hline & 2 & & & \end{array}$

Next, perform the first multiplication and summation:

1. Multiply the value of $a$ ($2$) by the first coefficient that was brought down ($2$). The result is $2 \times 2 = 4$.
2. Place this result ($4$) in the next column, directly below the second coefficient ($-5$).
3. Sum the values in that column: $-5 + 4 = -1$. This result is written on the bottom line.

Paso 3: Repeating the Iterative Process

The process of multiplication and summation is repeated for every remaining column until the end of the polynomial is reached.

For the next column:

1. Multiply the previous sum ($-1$) by $a$ ($2$): $-1 \times 2 = -2$.
2. Place the $-2$ below the third coefficient ($3$).
3. Sum the column: $3 + (-2) = 1$. Write this result on the bottom line.

For the final column:

1. Multiply the previous sum ($1$) by $a$ ($2$): $1 \times 2 = 2$.
2. Place the $2$ below the final independent term ($-7$).
3. Sum the column: $-7 + 2 = -5$. Write this final result on the bottom line.

The final row of the grid should look like this:

$\begin{array}{c|rrrr} & 2 & -5 & 3 & -7 \\ 2 & & 4 & -2 & 2 \\\hline & 2 & -1 & 1 & -5 \end{array}$

Paso 4: Interpretation of the Mathematical Result

The numbers resulting on the bottom line of the Ruffini grid provide the answer to the division:

• Quotient Coefficients: The numbers appearing before the final value ($2$, $-1$, and $1$) represent the coefficients of the quotient polynomial. Since the original polynomial was of degree 3 ($x^{3}$), the quotient will always be one degree lower, which is degree 2 ($x^{2}$). Therefore, the quotient is $C(x) = 2x^{2} - 1x + 1$.
• The Remainder: The very last number on the bottom line is the remainder of the division. In this example, the remainder is $-5$.

Final Result Expression

The relationship between the original polynomial, the divisor, the quotient, and the remainder can be expressed as:

$2x^{3} - 5x^{2} + 3x - 7 = (x - 2)(2x^{2} - x + 1) - 5$

Quick Summary Checklist

1. Extract the value of $a$ from the binomial $(x - a)$.
2. Write down the coefficients of the polynomial in descending order of degree.
3. Drop the leading coefficient to the bottom row.
4. Multiply the bottom-row value by $a$, place it in the next column, sum the column, and repeat until the table is complete.
5. The final number is the remainder; the other numbers form the coefficients of the new, lower-degree quotient polynomial.