The proof of Ruffini’s Theorem: definition, when and how the theorem is applied. Statement and explanation of an exercise.
Introduction to Ruffini’s Theorem
Ruffini’s Theorem is a theorem of fundamental importance in the decomposition of a polynomial of degree n. When it cannot apparently be decomposed into polynomials of lower degree by means of known rules. It is therefore the last technique to be used, almost a lifeline, in the decomposition of polynomials of any degree.
This method of decomposition was introduced in the 18th century by the mathematician Paolo Ruffini , who made it possible to decompose polynomials even when other techniques deriving from notable products failed.
This guide will provide the tools needed to understand this theorem and its proof.
Ruffini’s Theorem: When to Use It
Ruffini’s method is one of the safest systems for the decomposition of polynomials. Usually the system is applied whenever a polynomial of a sufficiently high degree must be decomposed and it is not possible to use the techniques relating to notable products. It is not advisable to use it for decomposable polynomials of the second degree, since the other techniques are certainly faster. If you find an alternative to Ruffini, always apply it, the theorem, in fact, although very effective, requires a procedure that is usually longer and more expensive also in terms of time.
Decomposition of polynomials
Very often we have to decompose a polynomial of degree n, but we do not always have a polynomial that can be decomposed based on notable products, as happens, for example, if we are faced with a difference or a sum of cubes. In this case we use Ruffini’s rule that allows us to decompose a polynomial of degree n into the product of a polynomial of degree n-1 and a polynomial of 1st degree.
It should be noted, however, that the rule does not work with all polynomials. In fact, the theorem is used only when the remainder is equal to 0, that is, only in the case in which the division is exact. To understand if the polynomial is divisible by a binomial and therefore if it is possible to apply the theorem, it is necessary to substitute the opposite value of a (that is, the number of the binomial with the sign changed) inside the polynomial in place of x.
For example: (3x² + 5x – 4) : (x – 2). Since a = -2 , I substitute its opposite, that is +2 in the polynomial and therefore: 3(2)2+5(2)-4=12+10-4= 18. Since the result obtained is different from zero then the division is not exact, the number identified in fact corresponds to the remainder.
Practical application of the theorem
Ruffini’s rule is a simplified version of the polynomial division algorithm that can be used only if the divisor is of the type x+b , that is, only if we have x+ a certain real number that can also have a negative value.
To understand concretely how to use the rule in the exercises, let’s imagine that we are asked to perform the division between two polynomials: A(x) = x3 – 4x – 2 and B(x) = x + 1.
We immediately notice that the divisor (x+1) appears as xa to which the real number 1 is added. To apply the theorem, it is necessary to construct a sort of table. At the top, in the first row, the coefficients of the dividend must be reported, that is, the numbers that appear before the powers of x of the polynomial A.
In the example we just gave, the coefficient of the second degree x is missing, in this case in the first row we will have to report the number 0. The minus two, however, is a known term of the dividend and therefore we will always put it in the first row of our table but divided with respect to the coefficients of x. Therefore, draw a vertical line after the -4 and before the -2. Remember that before transcribing the coefficients you will have to reorder them in order of degree.
At the bottom left we need to insert the opposite of the known term of the divisor (X+1). So at the bottom left we should write -1 (the opposite of the divisor).
At this point, once the table is fixed, we would have this situation: |1 0 -4 | -2-1|_ The first number, that is the 1, is brought down normally. The one must then be multiplied by the minus one and the result must be placed under zero. Now the zero must be added to the result of 1 for -1, that is -1, the result will be -1. Again we will have to multiply by -1 and add by -4 and so on. Remember that you multiply diagonally and add vertically.
The number that appears under the final -2 will be the remainder of the algorithm, while what you will have obtained before are the coefficients of the quotient. At this point you can rewrite the polynomial remembering that the degree will be one smaller than the initial one.
Returning to the example we would have: 1×2-1x-3=x2-x-3 remainder=1.
Proof of the theorem
Ruffini’s theorem is a necessary and sufficient condition for a polynomial P(x) to be divisible by a binomial of the type (xa) and that P(a) is equal to zero. Then for a polynomial P(X) and a binomial (xa), the relation P(x) = Q(x) * (xa) + R is valid, where R is a constant because the divisor is of the first degree.
Now P(x) takes values as a function of x; if x=a then the previous relation becomes: P(a) = Q(a) * (0) + R—> R=P (a). We have therefore given a definition of the remainder theorem: Given a polynomial P(x) and a binomial (xa) the remainder of the division of P(x) by (xa) is equal to P(a).
From the remainder theorem, Ruffini’s theorem follows. If we consider a polynomial N(x) and a binomial D(x) = x+a, the following properties hold: If N(-a) = 0 then N(x) is a multiple of D(x). If N(x) is a multiple of D(X) then N(-a)= 0.
Ruffini’s theorem provides a necessary and sufficient condition to state that a polynomial N(x) is a multiple of a binomial D(x), therefore R=0, therefore by the remainder theorem, N(-a)=0. For example: N(X) = X3-3X2+4.
It is easily verified that the number -1 is a root of the polynomial N(x), in fact: N(-1) =-1-3+4 = 0.
So by Ruffini’s theorem, the polynomial N(x) is a multiple of the binomial D(x) = x+1. Performing the long division we find Q(x)= x2-4x+4 and obviously R=0.
Therefore: N(x) = x3- 3×2+4 = D(x) * Q(X) = (X+1) (X2-4X+4).
In this example we have obtained a factorization of N(x). However, the proposed method does not suggest how to determine a root of N(x). In order to use Ruffini’s theorem to factor polynomials, it is good to keep in mind the following criterion for finding rational roots of a polynomial with integer coefficients. Any roots of a polynomial N(x) of degree n and integer coefficients must be found among fractions of the type +/- m/n, where m and n are submultiples respectively of the known term and the coefficient of the first degree term m of N(x).
