Cubics

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 ax^3+bx^2+cx+d, 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.

Factorising Cubics

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 3x^3-2x^2+5x have a common factor of x and so factor x  out to give x\left(3x^2-2x+5\right). 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 3x^2-2x+5 is (-2)^2-4\times 3\times 5=-56 which is negative and so 3x^2-2x+5 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 x=0 into the cubic expression to identify the y-intercept.
  • Next, factorise if possible and set y=0 to identify the roots. Note that, in y=x(x+1)^2 for example, x=-1 is a repeated root and the curve must touch the x-axis at x=-1.
  • Finally, place the graph on the axes so that all the above criteria are satisfied.

Examples

Factorise the following:

1. x^3+4x^2+3x
2. x^3-7x^2+10x
3. x^3-16x
4. 2x^3+7x^2-9x

Solution:

  1. x\left(x^2+4x+3\right)=x(x+3)(x+1)
  2. x\left(x^2-7x+10\right)=x(x-2)(x-5)
  3. x\left(x^2-16\right)=x(x+4)(x-4)
  4. x\left(2x^2+7x-9\right)=x(2x+9)(x-1)

1. Given that x=-2 is a root of the cubic x^3+x^2-x+2, factorise it completely.
2. Factorise f(x)=x^3-x^2-x+1 completely.

Solution:

  1. Since x=-2 is a root, (x+2) is a factor and factoring it out gives (x+2)(x^2-x+1) which can’t be factorised any further.
  2. By inspection, we can see that x=1 is a root of f(x) and so (x-1) is a factor. Using polynomial division or inspection we have f(x)=(x-1)(x^2-1) which factorises completely to f(x)=(x-1)(x-1)(x+1)=(x-1)^2(x+1).

Videos

https://youtu.be/PmKvbUlk7UQ

Factorising and sketching a simple cubic with a little coordinate geometry thrown in.

https://youtu.be/edBhGYKfY5w

Sketching and transforming a cubic mixed with a little differentiation.

Extra Resources

The following PDF has 9 cubic sketching exercises to complete. You will find the solutions to the exercises on the second page.

Open Cubic Sketching Exercise in New Window

Cubics

Here are the Cubics Questions by Topic. You will find solutions to these past exam questions at the back of the document.

Open Cubics Questions in New Window

CubicsExamQuestions

AS Maths Algebra & Functions

A2 Maths Algebra & Functions