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Cubic equations

If α, β, and γ are roots of equation ax3 + bx2 + cx + d = 0.
 
Then sum of roots α + β + γ = –b/a.
 
Sum of product taking two at a time = αβ + βγαγ = c/a. Product of all roots = αβγ = –d/a.

 

Notes:
  • A polynomial equation of degree n has n roots (real or imaginary).
  • If all the coefficients are real then the imaginary roots occur in pairs, i.e., number of imaginary roots is always even.
  • If the degree of a polynomial equation is odd then the number of real roots will also be odd. It follows that at least one of the roots will be real.

Solving Cubic Equation

By using factor theorem, together with some intelligent guessing, we can factorize polynomials of higher degree.
 
In summary, to solve a cubic equation of form ax3 + bx2 + cx + d = 0,
  1. Obtain one factor (xα) by trial and error.
  2. Factorize ax3 + bx2 + cx + d = 0 as (xα) × (hx2 + kx + s) = 0.
  3. Solve the quadratic expression for other roots.

Frequently used inequalities

  1. (xa)(xb) < 0 x (a, b), where a < b
  2. (xa)(xb) > 0 x (–∞, a) (b, ∞), where a < b
  3. x2a2 x [ –a, a]
  4. x2a2 x (–∞, –a] [a, ∞)
  5. If ax2 + bx + c < 0, (a > 0) x (α, β), where α, β (α < β) are roots of the equation ax2 + bx + c = 0
  6. If ax2 + bx + c > 0, (a > 0) x (–∞, α) (β, ∞), where α, β (α < β) are roots of the equation ax2 + bx + c = 0




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