If the polynomial f(x) can be divisible by the algebraic expression g(x), that is, a polynomial q(x) can be found so that f(x) = q (x) g(x), then g(x) is called a factor of f(x). A number can also be regarded as a factor.
Note: g(x)≠0, but q(x) can be equal to 0 (when f(x)=0).
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Factorization is closely related to solving higher-order equations. Linear equation and quadratic equation have relatively fixed and easy-to-master methods in junior high school. Mathematically, it can be proved that there are also fixed formulas for solving cubic equations and quartic equations. Just because the formula is too complicated, it will not be introduced in non-professional fields.
For factorization factors, cubic polynomials and quartic polynomials also have fixed decomposition methods, but they are more complicated. It has been proved that there is no fixed factorization method for general polynomials of degree five or above, and there is no fixed solution for univariate equations of degree five or above.
All univariate polynomials with more than three degrees can be decomposed in the real number range, and all univariate polynomials with more than two degrees can be decomposed in the complex number range.
This may seem a little incredible. Like x? +1, this is a univariate quartic polynomial, and it seems that it cannot be factorized. But its degree is higher than 3, so it must be factorized. It can also be decomposed by undetermined coefficient method, but the formula after decomposition is irregular.
This is because, from the basic theorem of algebra, there are always n roots in an unary polynomial of degree n, that is, an unary polynomial of degree n can always be decomposed into the product of n linear factors.
There is also a theorem: if the imaginary roots of real coefficient polynomials are paired with each other, the quadratic real coefficient factor can be obtained by multiplying the first factor corresponding to the imaginary roots of each pair of yokes. This conclusion holds.