New Zealand high school students learn about complex numbers in the final year of their secondary education. The complexity of the problems places the topic at Curriculum Level 8. However, existing time constrains limit a proper, systematic approach to this topic and I was compelled to create a set of alternative resources for enthusiastic students in order to help them successfully navigate through a variety of conceptual structures and obtain a more complete insight into complex polynomials and their factorisation.
This set contains illustrations of transitional quadratic realities. They visualise factorisation of quadratics with real coefficients, which have a negative discriminant and hence require a knowledge of complex numbers, in order to properly factorise them with the use of conjugate complex factors. The students are ready for this learning experience when they operate at Curriculum Level 8, after they have learned about surds (irrational roots of rational numbers), discriminant (b2 – 4 a c), squaring trinomials, the “completing the square” technique and complex numbers.
I have used the functions y = x2 + 2 x + c and y = x2 – 2 x + c, although (once the principles explained here are learned) one can use any similar quadratic function of this type (which has real coefficients but a negative discriminant).
This type of quadratic factorisation represents an entirely new learning experience, which is difficult to comprehend without geometric visualisations when learning this type of algebra. The approach is based on two principles applied one after another: completing the square and difference of two squares. Anyone who sees this representation will acknowledge that one picture means more than a thousand words!
When graphed, these quadratic functions generate mirrored parabolas growing out of the vertex of the corresponding rational parabolas, but rotated 90 degrees from the real x - y plane into the complex i x - y plane.
The greatest achievement when students follow this way is that they are now not only able to do quadratic factorisations with conjugate complex factors, but they can also explain what they do, how they do it and why they do it that way.
The sum of two squares is a frequently used concept in complex algebra. Little is known that it represents a special case (given separately in algebraic visualisations 8b) within a much bigger set of complex quadratics, visualised in set 8a.
This set effectively visualises factorisations of quadratics with an imaginary linear coefficient x2 ± i b x ± c, which, when factorised, produce complex factors (x ± i p) (x ± i q). To factorise any quadratic function y = x2 ± i b x ± c first we have to convert ± c into ± c i2 (having an opposite sign to c) and then the middle term i b x must be split into i p x + i q x, where p q = c and p + q = b. Only after splitting the middle term into two parts in this way a two - stage factorisation can be performed. The corresponding squares and rectangles (the geometric tools for visualisation) completely illustrate the whole process.
Each algebraic visualisation has its twin, where like terms are added, representing a simplified (or summarised) situation. Students can see that any chosen sequence represents a part of a greater continuum and it gives clues about the unseen reality.
The examples in set 8a elaborate the process of factorisation of complex quadratics of the form x2 ± i b x ± c. When a constant term has zero value (c = 0) the complex quadratic trinomial is being reduced into a quadratic binomial of the form x2 ± i b x and one of the factors is being reduced to a single x value (x ± 0).
The examples in set 8b separately elaborate the process of factorisation of complex quadratics obtained when i b x term has zero value (does not exist) and term c has a positive value. When this happens, a quadratic trinomial is being reduced into a quadratic binomial of the form x2 + c. The examples of this type fit into a group called the “sum of two squares”. This becomes understandable when we remember that any number representing the constant c also represents the squared value of its root. Because of that, the initial school examples at this level most often deal with c value representing a squared value of natural numbers (c = n2).
The examples in set 8c contain a selection of complex quadratics from a wider family x2 ± i b x – c (set 8a), which are transformed into x2 ± 2 i x y + (i y)2, when 2 y = b and (i y)2 = - c and can be factorised either as a squared complex sum or as a squared complex difference.
Based on the ideas which encompass rational complex numbers, I went a step further and incorporated complex surds into set 8d in order to complement the existing explanations and create a greater mathematical picture for factorising complex quadratics. This resulted in a challenging extension, which naturally belongs to the factorisation of this family of quadratic functions. It represents a real example of thinking outside the square in order to factorise any complex quadratic function of this type.
In this algebraic visualisation I have used quadratics x2 ± 2 i x + c although, once the principles explained here are learned, one can take any quadratic of this type and factorise it!
The algebraic situations used in set 8a to explain how to factorise a complex quadratic trinomial could be used for the complex quadratics factorisation
x2 – 2 i x + 3 into (x – 3 i) (x + i),
x2 – 2 i x + 8 into (x – 4 i) (x + 2 i) etc, or
x2 + 2 i x + 3 into (x + 3 i) (x – i),
x2 + 2 i x + 8 into (x + 4 i) (x – 2 i) etc,
but such a method of factorisation is ineffective when the constant c has a size of c = 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, … This approach to factorising complex quadratic trinomials (implemented in set 8d) is the only way to factorise quadratics x2 + 2 i x + c and x2 – 2 i x + c, which includes cases when c = 3, c = 8, c = 15, … c = (n2 – 1), n > 1, and therefore represents a new learning experience, which is rather difficult to understand without geometric visualisations when learning this kind of algebra. This approach is based on two principles applied one after another: completing the square and the difference of two squares. Anyone who sees this representation will acknowledge that one picture means more than a thousand words!
The examples in set 9a represent visualisations of factorisation of quadratics with complex coefficients x2 + (± b ± i c) x + i d which, when factorised, generate a product of rational and complex factors (x ± p) (x ± i q).
To factorise any complex quadratic of the type x2 + (± b ± i c) x ± i d, we first need to break the middle term (± b ± i c) x into the rational and complex components ± b x ± i c x, where b c = d. Only after splitting the middle term into the two parts and creating two pairs of terms can a two-stage factorisation be performed.
Again, we can choose how to group the four terms into the two pairs which then need to be factorised. One way is to create a pair of rational and pair of complex terms. Another way (which is more challenging for students and hence worthwhile to be visualised) is to create the two mixed pairs containing both rational and complex terms. Both ways of factorisation follow the same principle and generate the same final result. The only difference is the inverted order of the brackets, which represent an equivalent algebraic situation.
Each algebraic visualisation is followed by its twin, where like terms are added, representing a simplified (or summarised) situation. Students can see that any chosen sequence represents a part of a greater continuum and it gives clues about the unseen reality.
The examples in set 9b elaborate the process of factorisation of complex quadratics obtained when linear coefficients of x (b and i c) have an equal size, but opposite signs and complex quadratics become x2 + (y – i y) x – i y2 or x2 + (- y + i y) x – i y2.
The examples in set 9c contain complex quadratics x2 ± (y + i y) x + i y2 which can be factorised either as a squared complex sum or as a squared complex difference.
This set contains four challenging algebraic visualisations to help refine and polish students' thinking. The materials are made not only to educate but to create a space for imagination and investigation of the other possible mathematical pathways; they can be appealing to the learning needs of sophisticated mathematical thinkers at upper secondary levels. Mental visualisations and the understanding of complex quadratics opens up a space to explore avenues for the implementation of mathematics into contemporary physics.
The first set of visualisations (10a) represents the most general situation when expanding and factorising complex quadratics. It contains all the real and imaginary components which can be found in this process. All the other situations that can be encountered in mathematical publications for this level of education are simpler versions in which one or more components have zero value.
The second set of visualisations (10b) shows how to factorise another family of complex quadratics which, when factorised, give a combination of rational and irrational imaginary factors. It is given in two versions: the first one has reversed rational and imaginary components to complete the square easier. The second version has a standard ordering of rational and imaginary components, but here it is more difficult to see how to complete the square.
The third set of visualisations in this family of complex quadratics (10c) has been created to show how to factorise atypical complex quadratics containing only a real quadratic component and an imaginary constant term.
The fourth set of visualisations (10d) shows how to factorise complex quadratics with both components (a real rational and an imaginary irrational) present in both the linear and constant terms.
This (incomplete) set of algebraic visualisations has been made as a starter to encourage and motivate students (and their teachers) to begin to investigate the life of this mathematical rainforest which nurtures all possible combinations of rational and irrational, real and imaginary number components accompanying either some or all quadratic, linear and constant terms.
The principles that have already been learned, combined with new algebraic lessons (factor theorem) resulted in the creation of materials which additionally visualise a rich algebraic world. This set contains a selection of exemplars which follow the next idea: factorise the given polynomial, graph it initially as a non-factorised one and then as the factorised one, and compare the obtained graphs!
Exemplar 1 illustrates the factorisation of a positive cubic function which, when factorised, has 3 different factors and its roots are three different non – zero integers;
Exemplar 2 illustrates the factorisation of a positive cubic function which, when factorised, has 3 different factors and its roots are three different integers one of which is zero;
Exemplar 3 illustrates the factorisation of a positive cubic function which, when factorised, has 2 different factors (one is quadratic) and its roots are two different non – zero integers;
Exemplar 4 illustrates the factorisation of a positive cubic function which, when factorised, has 3 different factors and one of the roots is not an integer (it is a fraction or a decimal number);
Exemplar 5 illustrates the factorisation of a negative cubic function which, when factorised, has 3 different factors and its roots are three different non – zero integers;
Exemplar 6 illustrates the factorisation of a negative cubic function which, when factorised, has 3 different factors and its roots are three different integers, one of which is zero;
Exemplar 7 illustrates the factorisation of a negative cubic function which, when factorised, has 2 different factors (one is quadratic) and its roots are two different non – zero integers;
Exemplar 8 illustrates the factorisation of a negative cubic function which, when factorised, has 3 different factors and one of the roots is not an integer (it is a fraction or a decimal number);
Exemplar 9 illustrates the factorisation of a positive quartic function which, when factorised, has 4 different factors and its roots are four different integers;
Exemplar 10 illustrates the factorisation of a positive quartic function which, when factorised, has 3 different factors (one is quadratic) and its roots are three different integers;
Exemplar 11 illustrates the factorisation of a positive quartic function which, when factorised, has 2 different factors (one is cubic) and its roots are two different integers;
Exemplar 12 illustrates the factorisation of a positive quartic function which, when factorised, has only 1 (quartic) factor and its root is one integer;
Exemplar 13 illustrates the factorisation of a negative quartic function which, when factorised, has 4 different factors and its roots are four different integers;
Exemplar 14 illustrates the factorisation of a negative quartic function which, when factorised, has 3 different factors (one is quadratic) and its roots are three different integers;
Exemplar 15 illustrates the factorisation of a negative quartic function which, when factorised, has 2 different factors (one is cubic) and its roots are two different integers;
Exemplar 16 illustrates the factorisation of a negative quartic function which, when factorised, has only 1 factor (one is quartic) and its root is one integer.
This set contains a selection of exemplars representing the factorisation of polynomials which can only be performed with the use of complex factors.
Exemplar 1 represents the factorisation of quadratics with real coefficients, but (since the discriminant is negative), factorisation can be done with the use of conjugate complex factors.
Exemplar 2 represents the factorisation of quadratics with complex coefficients. A complete factorisation can only be accomplished with the use of complex factors. This particular example illustrates situations when one factor is real and another complex.
Exemplar 3 represents the factorisation of cubics with real coefficients, but (since the linear factor is real, and the quadratic factor also has real coefficients but a negative discriminant), a complete factorisation can only be accomplished with the use of conjugate complex factors.
Exemplar 4 represents the factorisation of cubics with complex coefficients. A complete factorisation can be accomplished only with the use of complex factors. This particular example illustrates situations when one factor is complex and two factors are real.
Exemplar 5 represents the factorisation of cubics with complex coefficients, and a complete factorisation can only be accomplished only with the use of complex factors. This particular example illustrates situations when two factors are complex and the third is real.
Exemplar 6 represents the factorisation of cubics with complex coefficients. A complete factorisation can only be accomplished with the use of complex factors. This particular example illustrates situations when all three factors are complex.
Expansion and factorisation principles have a wide application in other areas of Mathematics. In this set I have included several attractive examples for students’ enrichment and extension which utilise the learned principles in several elegant ways.