What are the types of polynomial operators

What are the laws of arithmetic and methods of polynomial multiplication?


The transformation of a polynomial into the form of a product of several simplest integers is called factoring that polynomial (also called factoring).


1. Common factorization method.

2. Formula method.

3. Group factorization method.

4. Method of rounding. [x^2+(a+b)x+ab=(x+a)(x+b)]

5. Combinatorial decomposition method.

6. Cross multiplication.

7. Double cross multiplication.

8. Matching method.

9. The method of removing and replacing terms.

10. Commutative method.

11. Long division method.

12. Root finding.

13. graphical method.

14. principal element method.

15. method of coefficients to be determined.

16. special value method.

17. factorization method.

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What is the polynomial formula?

The algorithms are as follows:

Addition and Multiplication

A finite sum of monomials is called a polynomial. The highest number of polynomials represented by sums of monomials of different classes in which the coefficients are not zero is called the number of this polynomial.

Addition of polynomials, is the addition of the coefficients of like terms in a polynomial, with the letters remaining the same (i.e., combining like terms). Multiplication of polynomials means combining like terms after multiplying each monomial in one polynomial with each monomial in another polynomial.

The set Fx{1,x2,…,xn}, formed by the totality of polynomials x1, x2,…,xn on F, becomes a ring for addition and multiplication of polynomials and is an integer ring with unit elements.

Multiple polynomials over a domain also have a factorization uniqueness theorem.

Division with Remainder

If f(x) and g(x) are two polynomials in F[x] and g(x) is not equal to 0, then there are unique polynomials q(x) and r(x) in F[x] that satisfy ƒ(x)=q(x)g(x)+r(x), where the number of r(x) is less than the number of g(x). At this point q(x) is called the quotient of g(x) divided by ƒ(x) and r(x) is called the remainder. When g(x) = x – α, then r(x) = ƒ(α) is called the cosine, where α is an element of F in the equation.

At this point division with remainder has the form ƒ(x)=q(x)(x-α)+ƒ(α), which is called the cosine theorem. g(x) is a sufficiently necessary condition for ƒ(x) to be a factor of ƒ(x) is that the remainder obtained by dividing g(x) by ƒ(x) equals zero. If g(x) is a factorization of ƒ(x), then it is also said that g(x) can divide ƒ(x), or that ƒ(x) is divisible by g(x). In particular, a sufficiently necessary condition for x-α to be a factorization of ƒ(x) is that ƒ(α) = 0, when α is said to be a root of ƒ(x).

If d(x) factors both ƒ(x) and g(x), then d(x) is said to be a common factor of ƒ(x) and g(x). If d(x) is a common factor of ƒ(x) and g(x), and either of the factors of ƒ(x) and g(x) is a factor of d(x), then d(x) is said to be a greatest common factor of ƒ(x) and g(x).

If ƒ(x)=0, then g(x) is a greatest common factor of ƒ(x) with g(x). When ƒ(x) and g(x) are all nonzero, you can apply the method of rolling over and dividing to find their greatest common factor.

Rollover division

Knowing that the two polynomials (x) and g(x) in the ring of monic polynomials F[x] are not equal to zero, dividing (x) by g(x) yields the quotient q1(x) and the remainder r1(x). If r1(x) = 0, then g(x) is a greatest common factor of (x) and g(x). If r1(x) ≠ 0, then dividing g(x) by r1(x) yields the quotient equation q2(x) and the remainder equation r2(x). If r2(x)=0,then r1 is a greatest common factor of (x) and g(x).

Otherwise, as the division continues, the number of remainders decreases, and after a finite s number of times, there must be zero remainders (i.e., zero polynomials) or zero remainders (i.e., zero polynomials). If the final residue results in a zero polynomial, then the original f(x) and g(x) are mutually prime; if the final residue results in a zero polynomial, then the greatest common factor of the original f(x) and g(x) is the last division with remainder is the divisor.

Using the algorithm of the tumbling division, the greatest common factor rs(x) of ƒ(x) and g(x) can be tabulated as a combination of ƒ(x) and g(x) whose coefficients are polynomials on F.

If the greatest common factor of ƒ(x) and g(x) is a zero degree polynomial, then ƒ(x) and g(x) are said to be mutually prime. Both the concepts of greatest common factor and reciprocity can be generalized to the case of several polynomials.

If ƒ(x), a polynomial in F[x] with a number not less than 1, cannot be expressed as the product of two polynomials in F[x] with lower numbers, then ƒ(x) is said to be an irreducible polynomial over F.

Any polynomial can be decomposed into a product of irreducible polynomials.

Functions of the form Pn(x)=a(n)x^n+a(n-1)x^(n-1)+…+a(1)x+a(0) are called polynomial functions, which are obtained by finitely many multiplications and additions of constants with the dependent variable x. Obviously, when n=1, its a primary function y=kx+b, and when n=2, its a quadratic function y=ax^2+bx+c.

Refer to Baidu Encyclopedia-Polynomials for the above content

What is the law of polynomials

Definition of polynomial: In mathematics, an algebraic formula consisting of the addition of several monomials is called a polynomial.

Each monomial in a polynomial is called a term of the polynomial, and the highest number of terms in these monomials is the number of times the polynomial is made. The terms in a polynomial that do not contain letters are called constant terms. In mathematics, a polynomial is an expression obtained from variables, coefficients, and the operations of addition, subtraction, multiplication, and powers (non-negative integer powers) between them.

Operational rules for polynomials

1. Multiplication of polynomials and polynomials

(1) When multiplying a polynomial by a polynomial, each term of one polynomial is multiplied by each term of the other polynomial, and then the products are added together.

(2) When multiplying two polynomials, you should prevent missing terms.

(3) A polynomial is a sum of monomials, each term including the preceding sign. During the operation, care should be taken to determine the sign of each term in the product.

2. Law of Multiplication of Monomials and Monomials

(1) Monomials and monomials are multiplied by their coefficients and powers of the same base, respectively. For letters contained in only one monomial, their exponents are used as a factor of the product.

(2) Steps in the operation of multiplying monomials and monomials

Multiply by their coefficients, including the sign; multiply by the power of the base; and leave unchanged the letters and their exponents that are contained in only one monomial. Take the product of these three parts as the result of the calculation.