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Equations and Symmetry P(X ) = 0 Where P Is a Polynomial

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Equations and Symmetry P(X ) = 0 Where P Is a Polynomial Equations and Symmetry P(X ) = 0 where P is a polynomial 2 I Quadratic equations: X + aX + b = 0 3 2 I Cubic equations: X + aX + bX + c = 0 4 3 2 I Quartic equations: X + aX + bX + cX + d = 0 5 4 3 2 I Quintic equations: X + aX + bX + cX + dX + e = 0 . and so on. Polynomial equation: 2 I Quadratic equations: X + aX + b = 0 3 2 I Cubic equations: X + aX + bX + c = 0 4 3 2 I Quartic equations: X + aX + bX + cX + d = 0 5 4 3 2 I Quintic equations: X + aX + bX + cX + dX + e = 0 . and so on. Polynomial equation: P(X ) = 0 where P is a polynomial 3 2 I Cubic equations: X + aX + bX + c = 0 4 3 2 I Quartic equations: X + aX + bX + cX + d = 0 5 4 3 2 I Quintic equations: X + aX + bX + cX + dX + e = 0 . and so on. Polynomial equation: P(X ) = 0 where P is a polynomial 2 I Quadratic equations: X + aX + b = 0 4 3 2 I Quartic equations: X + aX + bX + cX + d = 0 5 4 3 2 I Quintic equations: X + aX + bX + cX + dX + e = 0 . and so on. Polynomial equation: P(X ) = 0 where P is a polynomial 2 I Quadratic equations: X + aX + b = 0 3 2 I Cubic equations: X + aX + bX + c = 0 5 4 3 2 I Quintic equations: X + aX + bX + cX + dX + e = 0 . and so on. Polynomial equation: P(X ) = 0 where P is a polynomial 2 I Quadratic equations: X + aX + b = 0 3 2 I Cubic equations: X + aX + bX + c = 0 4 3 2 I Quartic equations: X + aX + bX + cX + d = 0 Polynomial equation: P(X ) = 0 where P is a polynomial 2 I Quadratic equations: X + aX + b = 0 3 2 I Cubic equations: X + aX + bX + c = 0 4 3 2 I Quartic equations: X + aX + bX + cX + d = 0 5 4 3 2 I Quintic equations: X + aX + bX + cX + dX + e = 0 . and so on. Quadratic equations: a method for solving was known to the Babylonians (ca. 2000 BCE) \Given the sum and product of two numbers, find the numbers." If p + q = A and pq = B, then p and q are the solutions of the equation X 2 − AX + B = 0 A bit of history a method for solving was known to the Babylonians (ca. 2000 BCE) \Given the sum and product of two numbers, find the numbers." If p + q = A and pq = B, then p and q are the solutions of the equation X 2 − AX + B = 0 A bit of history Quadratic equations: \Given the sum and product of two numbers, find the numbers." If p + q = A and pq = B, then p and q are the solutions of the equation X 2 − AX + B = 0 A bit of history Quadratic equations: a method for solving was known to the Babylonians (ca. 2000 BCE) If p + q = A and pq = B, then p and q are the solutions of the equation X 2 − AX + B = 0 A bit of history Quadratic equations: a method for solving was known to the Babylonians (ca. 2000 BCE) \Given the sum and product of two numbers, find the numbers." A bit of history Quadratic equations: a method for solving was known to the Babylonians (ca. 2000 BCE) \Given the sum and product of two numbers, find the numbers." If p + q = A and pq = B, then p and q are the solutions of the equation X 2 − AX + B = 0 particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: , Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE) , Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c These are \solutions by radicals"|that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) |that is, they express the solutions in terms of the coefficients using only the four algebraic operations and root extractions. Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J. Stillwell) These are \solutions by radicals" Cubic equations: particular equations were considered by Diophantus (ca. 250 CE), Omar Kayy´am(ca. 1100), Leonardo of Pisa (a.k.a. \Fibonacci," 1200) No general method of solution known Scipione del Ferro (ca. 1500): X 3 + bX = c Niccol`oTartaglia (1535): X 3 + aX 2 = c \The first clear advance in mathematics since the time of the Greeks." (J.
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