What is an algebraic equation?
Q: What is an algebraic equation?
A: An algebraic equation is an equation of the form P = Q, where P and Q are polynomials over a given field with one or more variables.
Q: How can two equations be equivalent?
A: Two equations are considered equivalent if they have the same set of solutions, meaning all solutions of one must also be solutions of the other and vice versa.
Q: What does it mean to solve an equation?
A: Solving an equation means finding the values for the variables that make the equation true. These values are called roots.
Q: Can algebraic equations over rational numbers always be converted to ones with integer coefficients?
A: Yes, by multiplying both sides by a number such as 42 = 2·3·7 and grouping terms in the first member, any algebraic equation over rational numbers can be converted to one with integer coefficients.
Q: When did ancient mathematicians want radical expressions for univariate equations?
A: Ancient mathematicians wanted radical expressions (like x=1+√5/2) for univariate equations (equations with one variable) during the Renaissance period.
Q: Who solved degree 3 and 4 equations during this time?
A: Gerolamo Cardano solved degree 3 equations and Lodovico Ferrari solved degree 4 equations during this time.
Q: Who proved that higher degree equations cannot always be solved using radicals?
A: Niels Henrik Abel proved in 1824 that higher degree equations cannot always be solved using radicals.
