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How to find the order of a polynomial

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The concepts of monomial and polynomial are often confused.

Let's see what is called a monomial and what is a polynomial. First of all, remember what was called the monomial in the lesson “Monomials”.

Note that the “inside” of the monomial (between the letters and the numerical coefficient) is only a multiplication sign. For example, in the monomial: 3ab = 3 · a · b

Polynomial called the algebraic sum of several monomials.

The monomials of which the polynomial is composed are called members of the polynomial.

Examples of polynomials: a + 2b 2 - c, 3t 5 - 4b, 4 - 6xy

It is easy to see that any polynomial consists of several monomials.

Consider the polynomial in more detail.

The question is why the algebraic sum of monomials is called a polynomial if there is a minus sign in the polynomial.

This is because in fact the “-” sign refers to the numerical coefficient of the monomial, which is to the right of the sign.

Any polynomial can be written according to the rule of signs as the sum of monomials.

In a polynomial, the sign to the left of the monomial refers to the numerical coefficient of the monomial itself.

Degrees of polynomials

PolynomialPower
polynomial
a 2 - 3a 2 b + x =
a 2 (monomial degree 2) - 3a 2 b(monomial degree 3) + x(monomial degree 1)
3
1
3
x 2 y 2 + 4x 2 =
1
3
x 2 y 2 (monomial degree 4) + 4x 2 (monomial degree 2)
4
8x 2 - 3a + 4 =
8x 2 (monomial degree 2) - 3a(monomial degree 1) + 4(monomial degree 0)
2

Any monomial is a polynomial. In fact, any monomial, in fact, is a polynomial that consists of only one monomial.

Examples of such polynomials: 2a 2 b, −3d 3, a.

Polynomial - the sum of several monomials in the expression

To grasp the essence of the term is very simple. If monomials are numbers, variables and degrees multiplied among themselves, then polynomials - these are monomials added to each other.

Consider an example.

  • The expressions 5 × 3 and 6ab, taken separately, will be considered monomials.
  • However, if you add them together - 5x 3 y + 6ab - then we get a polynomial expression.
  • Moreover, both parts of the polynomial will be its individual members.

A polynomial is any expression where monomials add together. But instead of a common name, more specific ones can be used. For example, an expression consisting of two such parts will be called a binomial, and of three, respectively, a trinomial.

Order is a polynomial

The order of the polynomials N (q) and M (q) is known.

To determine the order of a polynomial, starting from the highest degree, it is necessary to group its members in two So, for example, i - - x is a second-order polynomial, and 1 x - a third-order one.

As the order of the polynomial grows, the accuracy of the description grows, but at the same time, the interpretation of the model becomes more complicated - an analysis of the influence of each input. In addition, the more coefficients an equation contains, the more experiments it is necessary to find to find them: the minimum number of experiments is equal to the number of coefficients, and to be able to evaluate the adequacy, more experiments are needed than there will be coefficients. Equations of order higher than the third (with more than one argument) are rarely encountered in practice.

As the order of the polynomial grows, the accuracy of the description grows, but at the same time, the interpretation of the model becomes more complicated - an analysis of the influence of each input.

As the order of the polynomial grows, the accuracy of the description increases, however, the determination of the coefficients (b) becomes more complicated, since the more coefficients there are, the more experiments must be performed. Therefore, when creating statistical models, it is customary to use polynomials no higher than third order.

Let us denote the argument x and the order of the Taylor polynomial n with the capital letters X and N and we will enter them as the initial numeric data using the input command INPUT ii X, N in the first line of the program (INPUT means input in English), and the symbol ii is used here to clearly indicate the space between the word INPUT and the list of input values.

An appeal directive is introduced into the machine, in which it is indicated: N is the order of the polynomial, E is the specified accuracy, R1 is the digit capacity of the count, A is the array of multiples.

Since it is possible to significantly complicate the scheme when implementing functions (4.24) of high orders, the most important issue in the synthesis of FSS is the choice of the order of the Bessel polynomial.

The choice of cit carried out most often by the least squares method allows you to choose different filters. The order of the polynomial approximating the curve is set, and its parameters are determined by the least squares method so that the sum of the squared differences between the experimental points and the polynomial points is minimal. Smoothing is performed on 5 - 10 points with the help of curves of the second and third order. Digital filtering is used in many specialized gas chromatography computing tools.

In the general case of the matrix C, the coefficients ah are determined independently of each other. If we change the order of the regression polynomial (by adding or removing several terms), then all the coefficients ak will have to be determined anew, since the matrix C will change. If the experiment is planned so that the matrix C is diagonal, then ak will be determined independently from each other. Plans that have this property are called orthogonal.

The lack of derivatives on the left side indicates perfect transmission of input signals. The higher the order of the polynomial Q (p), the stronger, ceteris paribus, the form of the transmitted signals is distorted by the link.

To solve the problem, you must refer to the MULLER procedure. The meaning of the parameters of the procedure is as follows: N is the order of the polynomial, E is the specified accuracy, A is the name of the array of coefficients of the polynomial, U, V is the name of the arrays of real and imaginary parts of the roots of the polynomial.

Polynomials - what are the main actions they perform?

Polynomial expressions or expressions that include polynomials are quite long and complex. Therefore, it is recommended to simplify them as much as possible - from a non-standard view to lead to a standard one.

To write a polynomial in a standard form, it is necessary to perform all available actions with similar terms - add, subtract or multiply numerical coefficients, variables and degrees. The polynomial that can no longer be simplified even more will be considered standard.

For example, the expression 5x2 yy - 7xuh 2 + 5ax can be simplified to 5a 2 2x - 2x 3 y - and the resulting expression will be a standard polynomial. Writing it in an even simpler form is no longer possible.

There is also the concept of a degree of a polynomial.

  • It is simple to determine it - it is necessary to compare the degrees of each of the available terms.
  • The largest of them will be determined as the degree of the whole polynomial.
  • Moreover, the terms for which at first glance there is no degree are always considered as monomials in the first degree.

Often in problems there is a requirement to decompose the polynomial into separate factors. To do this is quite simple - you only need to determine whether, in principle, there is a common factor among the members of the expression. If there is one, it is put out of brackets, and the terms are left inside the brackets.

Properties of the minimum polynomial of a matrix

1. Any annihilating polynomial of a matrix is ​​divisible by a minimal polynomial (without remainder). In particular, the characteristic polynomial is divided by the minimal polynomial.

Indeed, suppose the contrary, let the annihilating polynomial [math] p ( lambda) [/ math] be divided by the minimal polynomial [math] mu _ ( lambda) [/ math] with the remainder:

2. For each square matrix [math] A [/ math], the minimal polynomial is unique.

In fact, if there were two minimal polynomials, then they would have the same degree and would be divided by each other, i.e. would differ only by a constant factor. However, the leading coefficients of these polynomials are equal to unity; therefore, such polynomials coincide.

3. All eigenvalues ​​of the matrix are the roots of the minimal polynomial.

Substituting into the equality (7.28) any root [math] lambda_i,

4. If the characteristic polynomial has the form (7.24), then the minimal polynomial of this matrix can be represented in the form

where [math] 1 leqslant m_1 leqslant n_1,

1 leqslant m_2 leqslant n_2 [/ math], etc., moreover, [math] m_1 + m_2 + ldots + m_k = m leqslant n [/ math].

This statement follows from property 3.

5. The minimum polynomial of the matrix [math] A [/ math] is found by the formula

where [math] d_( lambda) [/ math] is the largest common factor divisor of the (n-1) th minors of the characteristic matrix [math] (A- lambda E) [/ math].

Compare this equality with (7.27):

When dividing the λ-matrix [math] Delta_A ( lambda) cdot E [/ math] on the left by the characteristic matrix [math] (A- lambda E) [/ math] the quotient (left) must coincide due to the uniqueness of division. therefore

those. the polynomial [math] p ( lambda) [/ math] is the divisor of all elements of the adjoint matrix. Note that the degree of the polynomial [math] p ( lambda) [/ math] should be maximum, since the minimum polynomial [math] mu_A ( lambda) [/ math] has the smallest possible degree, and the sum of the degrees of these two polynomials in due to the equality [math] Delta_A ( lambda) = (- 1) ^ np ( lambda) mu_A ( lambda) [/ math] is fixed and equal to [math] n [/ math]. Therefore, the polynomial [math] p ( lambda) [/ math] is the largest common divisor of the elements of the adjoint matrix [math] (A- lambda E) ^ <+> [/ math]. Since the elements of the adjoint matrix are proportional to the minors of the (n-1) th order of the characteristic matrix, then [math] p ( lambda) = d_( lambda) [/ math].

Thus, [math] Delta_A ( lambda) = (- 1) ^ n cdot d_( lambda) cdot mu_A ( lambda) [/ math], whence formula (7.30) follows.

6. The minimum polynomial of the matrix [math] A [/ math] coincides with the last invariant factor [math] e_n ( lambda) [/ math] of the characteristic matrix [math] (A- lambda E) [/ math].

In fact, the greatest common factor is [math] d_( lambda) [/ math] the only minor of the nth order of the characteristic matrix [math] (A- lambda E) [/ math] differs from the determinant of this matrix by the factor [math] (- 1) ^ n [/ math], those. [math] Delta_A ( lambda) = (- 1) ^ n d_( lambda) [/ math]. Substituting this expression in (7.30), we obtain

Ways to find the minimum polynomial of a matrix

Let [math] A [/ math] be an nth-order square matrix. It is required to find its minimal polynomial.

1. Create the characteristic matrix [math] (A- lambda E) [/ math].

2. Bring it to the normal diagonal form [math] (A- lambda E) sim operatorname Bigl (e_1 ( lambda), e_2 ( lambda), ldots, e_n ( lambda) Bigr) [/ math].

The last invariant factor [math] e_n ( lambda) [/ math] is the minimal polynomial of the matrix [math] A [/ math] (by property 6).

1. Create the characteristic matrix [math] (A- lambda E) [/ math].

2. Find the characteristic polynomial [math] Delta_A ( lambda) = det (A- lambda E) [/ math].

3. Find the greatest common factor [math] d_( lambda) [/ math] minors (n-l) -ro of the order of the λ-matrix [math] (A- lambda E) [/ math].

4. Using the formula (7.30), obtain the minimal polynomial.

Example 7.12 Find the minimum polynomial of the matrix [math] A = begin1 & 1 & 1 1 & 1 & 1 1 & 1 & 1 end[/ math], using the minimal polynomial, find the degree [math] A ^ m [/ math] with the natural exponent [math] m in mathbb[/ math].

Decision. The first way. 1. We compose a characteristic matrix

2. We bring this λ-matrix to normal diagonal form. Swap the first and third lines. As the leading element, we select the unit that appears in the upper left corner of the matrix. Using the leading element, we make the remaining elements of the first row and first column equal to zero:

We take the [math] (- lambda) [/ math] element as the leading element and make all other elements of the second row and second column equal to zero. Then we multiply the second and third lines by (-1) so that the highest coefficients of the diagonal elements are equal to unity. We get the normal diagonal view:

The minimum polynomial of the matrix [math] mu_A ( lambda) = e_3 ( lambda) = lambda ^ 2-3 lambda [/ math].

The second way. 1. We compose the characteristic matrix (7.32).

2. We find the characteristic polynomial [math] Delta_A ( lambda) = 3 lambda ^ 2- lambda ^ 3 [/ math] (see Example 7.11).

3. We find the second-order minors of the characteristic matrix [math] (A- lambda E) [/ math]. We restrict ourselves to the minors located in the first two lines:

The expressions for the remaining minors coincide with those found. The greatest common divisor of the polynomials [math] lambda ^ 2-2 lambda, , (- lambda), , lambda [/ math] is [math] lambda [/ math], i.e. [math] d_2 ( lambda) = lambda [/ math].

4. By the formula (7.30) we obtain: [math] mu_A ( lambda) = frac <(- 1) ^ 3 (3 lambda ^ 2- lambda ^ 3)> < lambda> = lambda ^ 2 -3 lambda [/ math].

For verification, we calculate

Indeed, the minimal polynomial [math] mu_A ( lambda) [/ math] is annihilating, that is, [math] mu_A (A) = O [/ math]. Note that for the matrix [math] A [/ math], the minimal and characteristic polynomials differ only in the factor [math] (- lambda) [/ math].

We now find the degree [math] A ^ m [/ math] of the matrix [math] A [/ math]. To do this, consider the polynomial [math] lambda ^ m [/ math]. Divide it by the minimal polynomial [math] mu_A ( lambda) [/ math]. The remainder of division (a polynomial of degree no higher than the first) can be represented as [math] alpha , lambda + beta [/ math]. Get

where [math] p ( lambda) [/ math] is the quotient, and [math] ( alpha cdot lambda + beta) [/ math] is the remainder. We find the coefficients [math] alpha [/ math] and [math] beta [/ math], substituting the roots of the minimal polynomial into equality:

- when [math] lambda = 0 [/ math] we have: [math] 0 ^ m = p ( lambda) cdot0 + alpha cdot0 + beta [/ math],

- with [math] lambda = 3 [/ math] we have: [math] 3 ^ m = p ( lambda) cdot0 + alpha cdot3 + beta [/ math],

beta = 0 [/ math]. Therefore, [math] lambda ^ m = p ( lambda) ( lambda ^ 2-3 lambda) + 3 ^ lambda [/ math]. Now, substitute the variable [math] lambda [/ math] for the matrix [math] A: [/ math]

The result coincides with that obtained in example 7.10.

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