Theory of equations/Cubic and quartic equations

Cubic and quartic equations In previous chapters, we studied linear and quadratic equations and saw that every quadratic can be solved by completing the square, leading to the well‑known quadratic formula. In this section we move one step higher and discuss cubic (third‑degree) and quartic (fourth‑degree) equations. Historically, the solution of cubic and quartic equations in the 16th century marked a turning point in algebra. It led not only to explicit formulas for higher‑degree polynomials, but also to the discovery of complex numbers and to a deeper understanding of the structure of equations. Just as the Latin lesson starts with the nominative case as the “subject” of a sentence, we begin here by identifying the basic “subjects” of our study: the standard forms of cubic and quartic equations and the transformations that simplify them. Standard forms A general cubic equation in one variable is A general quartic equation (also called a biquadratic equation) is As with quadratics, the first step is usually to simplify these equations by scaling and shifting the variable. This makes later manipulations more transparent and reveals the underlying structure. Depressed cubic Any cubic can be reduced to a simpler form with no quadratic term, called a depressed cubic. Starting from we first divide by (assuming ) and obtain Now we remove the term by substituting After expanding and simplifying (a straightforward but somewhat lengthy algebraic step), the equation takes the form where This equation is called the depressed cubic associated with the original cubic. Depressed quartic Similarly, we transform a quartic equation into a depressed quartic without the cubic term. Divide by : Then substitute which eliminates the term and gives an equation of the form for certain coefficients that can be written explicitly in terms of . This is the standard starting point for the classical solution of quartics. Vocabulary Just as a Latin lesson introduces vocabulary, we collect here some key terms that will be used in this chapter. Core terminology Term Meaning cubic equation polynomial equation of degree 3, usually written quartic equation polynomial equation of degree 4, usually written depressed cubic cubic of the form (no term) depressed quartic quartic of the form (no term) discriminant quantity built from the coefficients of a polynomial that indicates whether roots are multiple or distinct resolvent auxiliary equation (often of lower degree) whose solution helps to express the roots of the original equation radicals expressions involving roots such as , used in closed‑form formulas Cardano’s formula classical formula expressing the solutions of a depressed cubic in terms of radicals Ferrari’s method classical method to solve quartic equations by reducing them to a resolvent cubic Key to Vocabulary: cubic = degree 3 polynomial equation quartic = degree 4 polynomial equation depressed = transformed to remove the next‑lower degree term (e.g. in a cubic) Overview of cubic equations In analogy with the way Latin grammar dissects the roles of nouns and adjectives, we now analyze cubic equations by examining their structure and the roles of the parameters and in the depressed form Number and nature of roots For quadratic equations, the discriminant tells us whether the roots are real and distinct, real and equal, or complex. For cubics, there is an analogous discriminant built from and : If , the cubic has three distinct real roots. If , at least two roots coincide (multiple root). If , the cubic has one real root and two non‑real complex conjugate roots. While the explicit expression for in terms of is more complicated, the principle is the same: the discriminant vanishes exactly when the polynomial has a repeated root. Cardano’s idea (without full derivation) The classical method, usually attributed to Cardano, solves the depressed cubic by seeking a solution of the form and imposing conditions on and so that the equation becomes solvable in radicals. Substituting , one obtains Expanding and collecting terms leads to If we choose and such that then the term multiplying disappears, and we are left with From we get Now let Then Thus and are roots of the quadratic equation Solving this quadratic gives explicit (though possibly complex) expressions for and , and then follow by choosing cube roots. Finally, yields a solution of the depressed cubic, and we return to to obtain solutions of the original cubic. The full Cardano formula is usually written compactly as Overview of quartic equations Quartic equations are more complicated than cubics, but they can still be solved in radicals. The general strategy, due to Ferrari, is to transform the depressed quartic into a form that can be factored as a product of two quadratics. Idea of Ferrari’s method (outline) The goal is to represent the quartic as where are chosen to match the coefficients . Expanding the right‑hand side gives Comparing coefficients with leads to the system Ferrari’s method cleverly introduces an auxiliary parameter and reduces the problem of finding to solving a suitable resolvent cubic. Once the quartic is factored into two quadratics, each quadratic can be solved by the usual quadratic formula. The explicit formulas are lengthy, so this chapter focuses on the structure of the method rather than on memorizing the final expression. Examples In the Latin material, short phrases illustrate how nouns and adjectives agree. Here we present short polynomial examples that illustrate the ideas above. Example 1: A simple cubic Consider This is already in depressed form with and . By inspection, we can factor it: So the roots are (double root) and . Note that the discriminant in this case is zero, consistent with the presence of a repeated root. Example 2: A biquadratic quartic Some quartics contain only even powers of , for example Setting reduces this to a quadratic which factors as Thus or , and therefore giving the four roots This special case shows that not every quartic requires the full power of Ferrari’s method. Example 3: From a general cubic to a depressed cubic Let Dividing by 2 gives We then substitute (since , and ). After expanding and simplifying, one obtains a depressed cubic of the form with explicit values of and . From that point, Cardano’s method can be applied. Exercises The following exercises mirror the pattern of the Latin lesson: first some direct manipulations, then translation‑style tasks where you convert between different “forms” of equations. Reduce each of the following cubic equations to depressed form by an appropriate change of variable. For each depressed cubic, identify , , and compute the discriminant . Decide how many real roots the equation has. Solve the following biquadratic equations by substituting . Write each of the following quartic equations in depressed form (eliminate the cubic term by substituting ). Check your work by expanding back the substituted form and verifying that the original equation is recovered. For each cubic in depressed form, compare your conclusion about the number of real roots with a sketch of the graph of . For each biquadratic equation, factor the corresponding quadratic in explicitly and then translate the result back into factors in . For one of the depressed quartics you obtained, attempt to factor it as and compare with the outline of Ferrari’s method given above.