Cubics are algebraic expressions of the third degree, meaning they contain a term with a variable raised to the power of three. They take the general form , where ‘a’ is non-zero and a, b, c, and d are constants. Cubics are integral to algebraic mathematics as they present more complex relationships than linear or quadratic equations.
Use Case: In the real world, cubics have numerous applications. For instance, they’re used in physics to model certain types of motion or energy potential. In engineering, cubics can describe certain properties of materials or the relationship between variables in electrical circuits. Economists use cubic functions to model various economic conditions, while in computer graphics, Cubic curves and surfaces play a crucial role in the generation of computer imagery.
Factorizing Cubic's
The most basic cubics questions might ask you to factorise a simple cubic where a factor of x can be taken out first. For instance, the terms in the expression have a common factor of x and so factor x out to give . You might think that this can be factorised further, however, in this case the quadratic cannot be factorised. This can be seen by noting that the discriminant of is which is negative and so has no roots. – see Discriminants. It follows that the cubic cannot be factorised further. In most cases, however, there will be some more factoring required. See Example 1.
In addition to the above, other cubics questions might involve factorising a more general cubic and may require knowledge of the factor theorem. See Example 2.
Sketching Cubics
Firstly, identify whether the cubic is positive or negative.
Then, substitute into the cubic expression to identify the -intercept.
Next, factorise if possible and set to identify the roots. Note that, in for example, is a repeated root and the curve must touch the x-axis at .
Finally, place the graph on the axes so that all the above criteria are satisfied
, where ‘a’ is non-zero and a, b, c, and d are constants. Cubics are integral to algebraic mathematics as they present more complex relationships than linear or quadratic equations.
have a common factor of x and so factor x out to give 
. You might think that this can be factorised further, however, in this case the quadratic cannot be factorised. This can be seen by noting that the discriminant of 
is 
which is negative and so 
into the cubic expression to identify the 
-intercept.
to identify the roots. Note that, in 
for example, 
is a repeated root and the curve must touch the x-axis at 
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