Tiêu đề 16 Theory of Equations.pdf ✅

MSTC 16: THEORY OF EQUATIONS 1. Vieta’s Formulas 1.1. Quadratic Equations For a quadratic equation ax 2 + bx + c = 0 with roots r1, r2, r1 + r2 = − b a r1r2 = c a 1.2. Cubic Equations For a cubic equation ax 3 + bx 2 + cx + d = 0 with roots r1, r2, r3, r1 + r2 + r3 = − b a r1r2 + r2r3 + r3r1 = c a r1r2r3 = − d a 2. Rational Root Theorem For a polynomial equation a0x n + a1x n−1 + a2x n−2 + ⋯ + an−1x + an = 0, the possible rational roots are: x = factors of ± an factors of ± a0 - Example: Find the possible rational roots of 3x 3 − x 2 + 7x + 6 = 0. [SOLUTION] The possible rational roots are: x = ±6, ±3, ±2, ±1 ±3, ±1 x = ± 1 3 , ± 2 3 , ±1, ±2, ±3, ±6 These values of x can be used as “guesses” for the possible roots of the equation. 3. Remainder Theorem For a polynomial function P(x), the remainder when P(x) is divided by x − a, the remainder is P(a). 4. Factor Theorem As an extension of the Remainder theorem, if P(a) = 0, then x − a is a P(x) factor and x = a is a root of P(x) = 0.