Ruffini's Rule

In the previous lesson, we explored how to divide two polynomials. In this lesson, we will learn a much faster method of dividing polynomials using a rule named after the Italian mathematician Paolo Ruffini: the Ruffini's Rule.

However, this rule for polynomial division can only be applied in cases where the divisor polynomial is a binomial of the form .

Ruffini's Rule

Ruffini's Rule is named after the Italian mathematician Paolo Ruffini who devised it in 1809.

Using this method, we can calculate the division between two polynomials very quickly, but it can be applied only when the divisor polynomial is a binomial of the type (where is a real number).

In reality, the procedure can also be applied when the divisor polynomial has a positive constant term. In fact, in this case, we can rewrite:

To understand how the rule works, let's start with an example.

Suppose we want to compute the division between the polynomials:

First, we need to arrange the dividend polynomial in decreasing order so that the term with the highest degree is the first and the one with the lowest degree is the last.

Next, we take the coefficients of the terms of the dividend polynomial and line them up in a row, then draw two vertical lines. The first vertical line is to the left of the coefficient of the term with the highest degree, while the second is to the right of the coefficient of the term with a degree of 1, separating the constant term from the rest.

Then, add an empty row and draw a horizontal line. Below is how it looks:

The second step involves adding, on the second row and to the left of the first vertical line, the constant term from the divisor polynomial but with the opposite sign:

Then, you pull down the first coefficient below the horizontal line. This will be the first coefficient of the resulting polynomial :

The third step involves multiplying the first coefficient we just found by the constant term of the divisor polynomial and adding it to the second coefficient on the top row. In our case, we need to multiply by and add the result to the second coefficient of the dividend, which is :

The value just found, , represents the coefficient of the second term of the result .

Now, we repeat the procedure with the term just found. So, we multiply the term by and add the result to the third coefficient of the dividend, which is :

The value just found, , represents the coefficient of the third term of the result .

The last step remains. Multiply the term by and add the result to the constant term of the dividend, which is :

This final value, , does not represent a coefficient of the resulting polynomial , but instead represents the remainder of the division between the two polynomials.

Result Form

The process is completed. What remains is to write the resulting polynomial and the remainder of the division.

To do this, it's important to remember that the resulting polynomial will have a degree equal to that of the dividend minus that of the divisor , as we saw in the lesson on polynomial division. Since the dividend has a degree of and the divisor has a degree of , the resulting polynomial will have a degree of . Therefore, we should write a second-degree polynomial with the coefficients we just found:

Similarly, the remainder will be:

which is a polynomial of degree zero.

Verification

Now that we have calculated the resulting polynomial and the remainder , we can verify that the division was executed correctly.

To do this, we need to multiply the resulting polynomial by the divisor polynomial and then add the result to :

As can be seen, the result is equal to the dividend . Thus, the division was executed correctly.

Examples

Let's clarify the Ruffini's Rule process with further examples.

Example

We want to divide the polynomial by the polynomial .

First, we set up the division table by inserting the coefficients and the constant term of the dividend polynomial, and inserting the opposite of the constant term of the divisor polynomial :

We bring down the first coefficient below the horizontal line:

We multiply the just-found first coefficient by the term and add the result to the second coefficient of the dividend :

The just-found value, , represents the coefficient of the second term of the resulting polynomial .

We repeat the process with the just-found term. So, we multiply the term by and add the result to the third coefficient of the dividend :

We conclude the process by multiplying the term by and adding the result to the constant term of the dividend :

Since the remainder is zero, this is an exact division. The resulting polynomial is:

Now, let's verify the just-found result. Multiply the resulting polynomial by the divisor polynomial :

The result matches the dividend , thus the division was executed correctly.

Example

We want to divide the polynomial by the polynomial .

Firstly, it should be noted that the dividend polynomial is not written in decreasing order of term degrees. Therefore, let's rewrite as:

Next, we set up the division table by entering coefficients and the constant term of the dividend polynomial and inserting the opposite of the constant term of the divisor:

We bring down the first coefficient below the horizontal line:

Multiply the just-found first coefficient by the term and add the result to the second coefficient of the dividend :

The just-found value, , represents the coefficient of the second term of the resulting polynomial .

Continue the procedure with the just-found term. So, multiply the term by and add the result to the third coefficient of the dividend :

Finally, multiply the term by and add the result to the constant term of the dividend :

The remainder is zero, while the quotient is:

Let's verify that the result is correct. Multiply the resulting polynomial by the divisor polynomial . We should have:

The result is correct.

Example

We divide the polynomial:

by the polynomial:

In this case, we need to observe that is not a complete polynomial. In other words, it lacks a term, specifically the term of degree 2. When preparing the table for division, we need to insert the missing coefficients using the value :

From here, we can proceed as usual. Bring down the first coefficient below the horizontal line:

Multiply this value by the term and add the result to the second coefficient of the dividend:

Continue with the next term:

Lastly, multiply the term by and add the result to the constant term of the dividend:

We have found that the quotient is:

and the remainder is:

Let's verify that the result is correct. It should be:

Therefore:

The result is correct.

In Summary

Through Ruffini's rule, we've seen how we can quickly divide one polynomial by another. This rule is valid as long as the divisor polynomial is of the first degree and has the form:

Ruffini's rule, besides being a division technique, also allows us to obtain fundamental results for polynomial theory. In the next lesson, we will see its very important first application: the Remainder Theorem of Polynomial Division.

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