Relationships between the roots and coefficients of a cubic equation

Relationships between the roots and coefficients of a cubic equation

Relationships Between Roots and Coefficients of Cubic Equations

General Form of Cubic Equations

  • A cubic equation generally has the form: ax³ + bx² + cx + d = 0, where ‘a’ is the coefficient of the cubic term, ‘b’ is the coefficient of the quadratic term, ‘c’ is the coefficient of the linear term and ‘d’ is the constant term.
  • The cubic equation is said to have three roots which can be real or complex. They are usually represented as α, β, and γ.

Sum of the Roots

  • The sum of all three roots α, β, and γ of the cubic equation ax³ + bx² + cx + d = 0 is equal to -b/a.
  • This means if you add the three roots together, the result should be negative of the coefficient of the square term divided by coefficient of the cube term.

Product of the Roots

  • The product of all three roots α, β, and γ of the cubic equation ax³ + bx² + cx + d = 0 is equal to d/a.
  • The roots multiply to give the result of the constant term divided by the coefficient of the cube term.

Sum of the Product of the Roots Taken Two at a Time

  • The sum of the product of roots taken two at a time (αβ + αγ + βγ) from the cubic equation ax³ + bx² + cx + d = 0 equals to c/a.
  • This relation means if you multiply each pair of roots and add these products together, the result should be equal to coefficient of the linear term divided by coefficient of the cubic term.

Understanding Roots in Different Conditions

  • It’s possible for cubic equations to have one, two, or three real roots, or a mix of real and complex roots.
  • This depends upon the discriminant of the cubic equation.
  • Without getting too deep into the discriminant, remember that it can help you determine the nature of the roots of the equations similar to how it works in quadratic equations.

Complex Roots in Cubic Equations

  • For cubic equations with real coefficients, if there are any complex roots, they must come as conjugate pairs.
  • This is due to the nature of polynomial equations with real coefficients.

Relationship Between Graphs and Roots

  • The x-intercepts of the graph of a cubic function correspond to the real roots of the cubic equation.
  • If a cubic equation has three distinct real roots, then the graph will cross the x-axis at three points. If there are repeated roots, then it may only cross at one or two points.