This problem can be solved by big cross multiplication.

Which is the binary multiplication.

First put the formula of X2+7xy- 18y2.

Get (-3X+9Y)(8X-2Y)

And then match X.

Get (X+9Y+8)(X-2Y-3)

I'll give you a double cross explanation.

On the draft paper, A is decomposed into mn product as one column, C into pq product as the second column, and F into jk product as the third column. If MQ+NP = B, PK+QJ = E, MK+NJ = D, it is the first one. Then the original formula = (MX+py+j) (NX+QY+k)

Example: 3x2+5xy-2y2+x+9y-4 = (x+2y-1) (3x-y+4) because 3 = 1× 3, -2 = 2× (- 1),-.

Migration of double cross multiplication

Decomposition of quadratic quintic formula

Key: Take the missing item as the coefficient 0, and multiply 0 by any number to get 0, for example: AB+B 2+A-B-2 = 0×/Kloc-0 /× A 2+AB+B 2+A-B-2 = (0× A+B+1) ()

Decompose quartic quintic formula

Tip: Let x 2 = y, and divide cx^2 into the sum of mx^2 and ny by decomposition method. For example: 2x4+13x3+20x2+1x+2 = 2y2+13xy+15x2+5y+1x. +1) (2x+1) (x2+5x+2) In short, we often use cross multiplication when we decompose quadratic trinomials. For some binary quadratic sextuples (ax 2+bxy+cy 2+dx+ey+f), we can also use cross multiplication to decompose the factors. The factorization factor is 2x 2-7xy-22y 2-5x+35y-3. We arrange the above formula in descending order of X and take Y as a constant, then the above formula can be transformed into 2x 2-(5+7y) x-(22y 2-35y+3), which can be regarded as a quadratic trinomial about X and decomposed into-22Y2+35y-3 = (2y-3) (-/kloc-). Then the quadratic trinomial about x is decomposed by cross multiplication, so the original formula is = [x+(2y-3)] [2x+(-66). (2x- 1 1y+ 1)。 (x+2y)(2x- 1 1y)= 2 x2-7xy-22 y2； (x-3)(2x+ 1)= 2 x2-5x-3； (2y-3)(- 1 1y+ 1)=-22y 2+35y-3。 This is called double cross multiplication. The steps of factorizing the polynomial AX 2+Bxy+CY 2+DX+EY+F with the binary multiplication factor are as follows. (2) The constant term f is decomposed into two factors and filled in the third column, and the sum of cross products formed by the second column and the third column is required to be equal to ey in the original formula. The sum of the cross products formed by the first column and the third column is equal to dx.2 For the root method, we call the algebraic expression in the form of anxn+an-1xn-1+…+a1x+A0 (n is a non-negative integer) as the unary polynomial about x, and use f(x) and g.g. F (-2) = (-2) 2-3× (-2)+2 = 12。 If f(a)=0, then a is called the root of polynomial f(x). Theorem 1 (factorial theorem) If A is polynomial f(x), the key to finding the first factorial of unary polynomial f(x) is to find the root of polynomial f(x). For any polynomial f(x), there is no general method to find the root. But when all the coefficients of polynomial f(x) are integers, that is, integer coefficient polynomials, the following theorem is often used to determine whether it has a rational root.

I wish you progress in your study!