* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Important Theorems for Algebra II and/or Pre
Survey
Document related concepts
Georg Cantor's first set theory article wikipedia , lookup
History of trigonometry wikipedia , lookup
System of polynomial equations wikipedia , lookup
Elementary mathematics wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Fermat's Last Theorem wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Nyquist–Shannon sampling theorem wikipedia , lookup
Central limit theorem wikipedia , lookup
Factorization wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Transcript
Important Theorems for Algebra II and/or Pre-Calculus State the following theorems and provide an illustration of each theorem. Do each theorem and an illustration/example on exactly one side of a sheet of notebook paper. You may use both the front side and backside of the notebook paper, but only one theorem may be completed per side. Your illustration/example should be brief and to the point, not just one part in the solution of a long problem. 1. 2. 3. 4. 5. 6. 7. 8. Fundamental Theorem of Algebra (FTA) Rational Root Theorem (RRT) Complex Conjugate Root Theorem (CCRT) Factor Theorem Remainder Theorem Discriminant Theorem Descartes' Rule of Signs Sum and Product of Roots Theorem ← called Nature of Solutions ← Theorem 5 in our text - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Since these theorems are worded differently, and often poorly, in different texts, I am providing you with my preferred version of each theorem. You may cite a different version if you prefer. 1. Fundamental Theorem of Algebra (FTA) Part (a) : Statement of the Theorem Fundamental Theorem of Algebra (FTA) Every polynomial equation with positive degree n has exactly n complex roots. Note that for this theorem, a double root counts as 2 roots, a triple root counts as three roots, and so on. Part (b) : Illustration/Example By the FTA, the equation x 3 = 8 has 3 solutions (over the complex numbers of course). 2. Rational Root Theorem (RRT) Part (a) : Statement of the Theorem Rational Root Theorem (RRT) bg Let P x be a polynomial with integral coefficients and a nonzero constant term. b g = 0 has a rational root p that is in lowest terms, q then p must be an integral factor of the constant term, and q must be an integral If the polynomial equation P x factor of the leading coefficient. Part (b) : Illustration/Example (for you to do) 3. Complex Conjugate Root Theorem (CCRT) Part (a) : Statement of the Theorem Complex Conjugate Root Theorem (CRRT) If a polynomial equation with real coefficients has a + bi as an imaginary root, then a − bi is also a root. Part (b) : Illustration/Example (for you to do) 4. Factor Theorem Part (a) : Statement of the Theorem Factor Theorem bg b g = 0, For the polynomial P x and the polynomial equation P x x − r is a factor if and only if r is a root Part (b) : Illustration/Example (for you to do) 5. Remainder Theorem Part (a) : Statement of the Theorem Remainder Theorem bg bg When a polynomial P x is divided by x − a , the remainder is P a . Part (b) : Illustration/Example (for you to do) 6. Discriminant Theorem Part (a) : Statement of the Theorem Discriminant Theorem Given the quadratic equation ax + bx + c = 0 , where a , b , and c are real numbers: 2 If b − 4ac > 0 , there are two unequal real roots. 2 If b − 4ac < 0 , there are two unequal imaginary roots (called complex conjugates). 2 If b − 4ac = 0 , there is one real root (called a double root). 2 Part (b) : Illustration/Example (for you to do) 7. Descartes' Rule of Signs Part (a) : Statement of the Theorem Descartes' Rule of Signs bg If P x is a polynomial with real coefficients, then 1. bg=0 the number of positive roots of P x bg is either equal to the number of variations of sign of P x or is less than this number by a positive even integer ; 2. bg=0 the number of negative roots of P x b g is either equal to the number of variations of sign of P − x or is less than this number by a positive even integer . Part (b) : Illustration/Example Here's a good equation to say something immediate about: 8. x 4 + 2 x 3 − 2 x 2 − 6x − 3 = 0 Sum and Product of Roots Theorem Part (a) : Statement of the Theorem Sum and Product of Roots Theorem For the equation an x + an −1 x n n −1 + an − 2 x n − 2 + K + a1 x + a0 = 0 , with an ≠ 0 : an − 1 the sum of the roots is − the product of the roots is R| a |S a || − a T a an 0 n 0 n Part (b) : Illustration/Example (for you to do) if n is even if n is odd