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Discriminant of a Cubic Equation


In the Algebra 1 section it was shown how to work out the discriminant of a quadratic equation, and how it can be useful to do so.



ax^2 + bx + c \space \space \space , \space \space \space \triangle = b^2 \space {\text{–}} \space 4ac


The discriminant  \triangle  could be positive, negative or zero.

With the result telling you about the type of roots/zeros the quadratic had.


–   b^2 \space {\text{–}} \space 4ac \space > \space 0   ,   real distinct roots
–   b^2 \space {\text{–}} \space 4ac \space = \space 0   ,   real equal roots
–   b^2 \space {\text{–}} \space 4ac \space < \space 0   ,   complex roots


Things work the same way with the discriminant of a cubic equation, but it takes a bit more work than the quadratic case.



Discriminant of a Cubic Formula:


For a standard cubic equation,    ax^3 + bx^2 + cx + d = 0.

The discriminant is given by the following.

\space {\triangle}_3 \space = \space b^{\tt{2}}c^{\tt{2}} \space {\text{–}} \space 4ac^{\tt{3}} \space {\text{–}} \space 4b^{\tt{3}}d \space {\text{–}} \space 27a^{\tt{2}}b^{\tt{2}} + 18abcd


A cubic equation of degree  3  will have at most  3  roots/solutions, which will be one of the following.

If  {\triangle}_3 > 0,   3 distinct real roots/zeros.

If  {\triangle}_3 = 0,   3 real roots/zeros, of which 2 are equal.

If  {\triangle}_3 < 0,   1 real root/zero, and 2 complex roots which are a conjugate pair.





Discriminant of a Cubic Equation
Examples



(1.1) 

Determine the nature of the zeros/roots of the cubic equation,    x^3 + 4x^2 + 2x \space {\text{–}} \space 1.

Solution   

a = 1 \space , \space b = 4 \space , \space c = 2 \space , \space d = {\text{-}}1

Using the formula.

{\triangle}_3 \space = \space b^{\tt{2}}c^{\tt{2}} \space {\text{–}} \space 4ac^{\tt{3}} \space {\text{–}} \space 4b^{\tt{3}}d \space {\text{–}} \space 27a^{\tt{2}}b^{\tt{2}} + 18abcd

{\triangle}_3 \space = \space (4)^{\tt{2}}(2)^{\tt{2}} \space {\text{–}} \space 4(1)(2)^{\tt{3}} \space {\text{–}} \space 4(4)^{\tt{3}}({\text{-}}1) \space {\text{–}} \space 27(1)^{\tt{2}}(4)^{\tt{2}} + 18(1)(4)(2)({\text{-}}1)

= \space 64 \space {\text{–}} \space 32 + 256 \space {\text{–}} \space 27 \space {\text{–}} \space 154 \space = \space 107

The graph of this polynomial is shown below, and as expected from the discriminant result, there are  3  distinct real roots/zeros.

The discriminant of a cubic tells us what type of roots to expect, and we can see them on the polynomial graph.





(1.2) 

Determine the nature of the zeros/roots of the cubic equation,    x^3 + x^2 \space {\text{–}} \space 8x \space {\text{–}} \space 12.

Solution   

a = 1 \space , \space b = 1 \space , \space c = {\text{-}}8 \space , \space d = {\text{-}}12

{\triangle}_3 \space = \space b^{\tt{2}}c^{\tt{2}} \space {\text{–}} \space 4ac^{\tt{3}} \space {\text{–}} \space 4b^{\tt{3}}d \space {\text{–}} \space 27a^{\tt{2}}b^{\tt{2}} + 18abcd

{\triangle}_3 \space = \space (1)^{\tt{2}}({\text{-}}8)^{\tt{2}} \space {\text{–}} \space 4(1)({\text{-}}8)^{\tt{3}} \space {\text{–}} \space 4(1)^{\tt{3}}({\text{-}}12) \space {\text{–}} \space 27(1)^{\tt{2}}({\text{-}}12)^{\tt{2}} + 18(1)(1)({\text{-}}8)({\text{-}}12)

= \space 64 + 2048 + 48 \space {\text{–}} \space 3888 + 1728 \space = \space 0

As we see in the polynomials graph below, there are  3  real roots, of which  2  are equal in the negative direction of the x-axis.

Graph with 3 real roots, 2 of them equal.





(1.3) 

Determine the nature of the zeros/roots of the cubic equation,    x^3 \space {\text{–}} \space 4x^2 \space {\text{–}} \space 4x \space {\text{–}} \space 16.

Solution   

a = 1 \space , \space b = {\text{-}}4 \space , \space c = {\text{-}}4 \space , \space d = {\text{-}}16

{\triangle}_3 \space = \space b^{\tt{2}}c^{\tt{2}} \space {\text{–}} \space 4ac^{\tt{3}} \space {\text{–}} \space 4b^{\tt{3}}d \space {\text{–}} \space 27a^{\tt{2}}b^{\tt{2}} + 18abcd

{\triangle}_3 \space = \space ({\text{-}}4)^{\tt{2}}({\text{-}}4)^{\tt{2}} \space {\text{–}} \space 4(1)({\text{-}}4)^{\tt{3}} \space {\text{–}} \space 4({\text{-}}4)^{\tt{3}}({\text{-}}16) \space {\text{–}} \space 27(1)^{\tt{2}}({\text{-}}16)^{\tt{2}} + 18(1)({\text{-}}4)({\text{-}}4)({\text{-}}16)

= \space 256 + 256 \space {\text{–}} \space 4096 \space {\text{–}} \space 6912 \space {\text{–}} \space 4608 \space = \space {\text{-}}15104

A negative discriminant value tells us that there is just one real root on the x-axis.

Polynomial graph with 1 real root and complex roots.






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