Vieta Theorem The French mathematician Veda first discovered this relationship between the roots and coefficients of algebraic equations, so people call this relationship Vieta Theorem. History is very interesting. This theorem was obtained by David in16th century. The proof of this theorem depends on the basic theorem of algebra, but Gauss first demonstrated it in 1799. It can be inferred from the basic theorem of algebra that any unary equation of degree n must have roots in a complex set. So the left end of the equation can be decomposed into the product of linear factors in the range of complex numbers: where is the root of the equation. Vieta's theorem is obtained by comparing the coefficients at both ends. Vieta's theorem AX2+BX+C=0 X 1 and X2 are a skill in the application of the equation x1+x2 =-b/ax1* x2 = c/aa Vieta theorem. If Vieta's theorem is combined with the decomposition formula α β (α+β)+/kloc-0, For example, P+Q = 198 is known. Find the integer root of the equation x2+px+q = 0. Solution of' 94 Zu Chongzhi Cup Mathematical Invitational Tournament: Let the two integer roots of the equation be x 1 and X2, and let X 1 ≤ X2. According to Vieta's theorem, X 1+X2 =-P, x1x2 = Q. So x1x2-(x1+x2) = p+q =198, that is, x/kloc. X 1 =- 198，x2 = 0。 Example 2 It is known that the two roots of the equation x2-( 12-m) x+m- 1 = 0 are positive integers, so find the value of m.. Solution: Let two positive integer roots of the equation be X65438+. And suppose x 1 ≤ x2. According to Vieta theorem, x 1+x2 = 12-m, x 1x2 = m- 1. So x1x2+x1+x2 = 65438+. X 1 = 2，x2 = 3。 Therefore, there is m = 6 or 7. Example 3 is to find the number k, so that the roots of the equation kx2+(k+1) x+(k-1) = 0 are all integers. Solution: If k = 0, x = 65438. Vieta theorem ∴ x 1x2-x 1-x2 = 2, (x1-kloc-0/) (x2-1) = 3. Because x 1- 1, x2-. And α > 1 > β, verification: P+Q > 1. (97 Junior Middle School Mathematics Competition in Sichuan Province) It is proved that the two roots of the equation -x2+px+Q = 0 are α and β. According to Vieta's theorem, α+β = P, α β =-Q. So P.