Cubic and quartic equations

Section: Cubic and Quartic Equations

• A cubic equation is a polynomial equation of degree 3, with the general form f(x) = axe^3 + bx^2 + cx + d = 0. Here, the coefficients a,b,c,d are real numbers with a ≠ 0.
• A quartic equation is another type of polynomial equation, but of degree 4: it takes on the general form f(x) = axe^4 + bx^3 + cx^2 + dx + e = 0. The coefficients a,b,c,d,e are real or complex numbers with a ≠ 0.
• Cubic equations can be solved by a method similar to the quadratic formula, known as Cardano’s method. This method involves some introducing intermediate variable to simplify the equation and then using factorization or the formula for solving quadratic equations.
• A quartic equation can be solved using Ferrari’s method or Descartes’ method. However, these methods are complex: a simpler approach is to reduce the quartic to a quadratic, if possible.
• The solutions (roots) of cubic and quartic equations can be real or complex. In other words, they can fall within the set of real numbers or involve a combination of real and imaginary numbers.
• Similar to the quadratic case, the nature of the roots of a cubic or quartic equation can be determined without actually solving the equation. For a cubic equation, if the discriminant is positive, there are three real roots; if it is zero, there is a repeated real root; if it’s negative, there is one real root and two complex roots. For a quartic, the nature of roots is determined by a slightly more complex criterion.
• A cubic polynomial will always touch/intersect its axis at least once, as it ranges from negative to positive infinity and vice versa.
• A quartic polynomial can touch/intersect its axis up to four times, because it has a degree of 4.
• Both cubic and quartic polynomials may have turning points (local maxima or minima). A cubic has at most two turning points, while a quartic can have up to three.
• The process of graphing these types of functions involves identifying key features such as y-intercepts, x-intercepts, turning points, and points of inflexion, then plotting accordingly.
• Remember to practise solving these types of equations, as well as related word problems, as part of your revision.