VISUAL ALGEBRA III represents a set of systematised factorisations of polynomials and extension materials consisting of 48 algebraic visualisations created to give meaning to:

• 7 Factorisations of quadratics with real coefficients and conjugate complex factors of the following algebraic forms:
  • x² ± 2 x + c,   [c = 1, 2, 3, 4, 5, 6, 7, 8, 9];
• 8a Factorisations of quadratics with an imaginary linear term and both complex factors of the following algebraic forms:
  • x² + i b x + c
  • x² + i b x – c
  • x² – i b x + c
  • x² – i b x – c
  • x² + i b x
  • x² – i b x;
• 8b Sum of two squares' complex quadratic factorisations of the form:
  • x² + y²,   [y = 1, 2, 3, 4, 5, 6];
• 8c Full square complex quadratic factorisations of the following algebraic forms:
  • x² + 2 i x y – y²   and   x² – 2 i x y – y²,   [y = 1, 2, 3, 4, 5, 6];
• 8d Complex quadratic factorisations using surds of the following algebraic forms:
  • x² + 2 i x + c   and   x² – 2 i x + c,   [c = 1, 2, 3, 4, 5, 6, 7, 8, 9];
• 9a Factorisations of quadratics with a complex linear and an imaginary constant term and, when factorised, with a combination of rational and complex factors of the following forms:
  • x² + (± b ± c i) x ± d i
  • x² ± b x
  • x² ± i c x;
• 9b Factorisations of complex quadratics which, when factorised, generate rational and complex factors with opposite signs:
  • x² + (y – i y) x – i y²,   [y = 1, 2, 3, 4, 5, 6]
  • x² + (- y + i y) x – i y²,   [y = 1, 2, 3, 4, 5, 6];
• 9c Factorisations of complex quadratics which, when factorised, generate equal rational and complex factors::
  • x² + (y + i y) x + i y²,   [y = 1, 2, 3, 4, 5, 6]
  • x² + (- y – i y) x + i y²,   [y = 1, 2, 3, 4, 5, 6];

Factorisations of other types of complex quadratics such as:

• 10a The most general situation when expanding and factorising complex quadratics;
• 10b Factorisations of complex quadratics with irrational complex factors;
• 10c Factorisations of complex quadratics containing only a real quadratic component and an imaginary constant term;
• 10d Factorisations of complex quadratics containing both components (a real rational and an imaginary irrational) in both the linear and constant terms;

Exemplars of factorisations of rational polynomials:

• 11a 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;
• 11a 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;
• 11a 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;
• 11a 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);
• 11a 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;
• 11a 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;
• 11a 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;
• 11a 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);
• 11a Exemplar 9 illustrates the factorisation of a positive quartic function which, when factorised, has 4 different factors and its roots are four different integers;
• 11a 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;
• 11a 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;
• 11a Exemplar 12 illustrates the factorisation of a positive quartic function which, when factorised, has only 1 (quartic) factor and its root is one integer;
• 11a Exemplar 13 illustrates the factorisation of a negative quartic function which, when factorised, has 4 different factors and its roots are four different integers;
• 11a 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;
• 11a 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;
• 11a 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;

Exemplars of factorisations of polynomials with complex factors:

• 11b 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;
• 11b Exemplar 2 represents the factorisation of quadratics with complex coefficients. A complete factorisation can only be accomplished with the use of complex factors. This exemplar illustrates situations when one factor is real and another complex;
• 11b 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;
• 11b 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;
• 11b 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;
• 11b 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;

Extension exercises:

• 12a   7 ^{2 x} + 7 ^x = 20;
• 12b   6 ^{2 x} – 6 ^x = 56;
• 12c   3 ^{2 x} – 3 ^{x + 1} = 4;
• 12d   4 ^x – 4 ^{3 – x} = 12;
• 12e   5 ^{2 x – 3} = 2 * 5 ^{x – 2} + 3;
• 12f   9 ^x = 3 ^{(x + 1) / x};
• 12g   2 ^{(x + 1) / x} * 0.5 ^{x + 1} = 1;
• 12h   0.5 ^{x^2 – 20 x + 59.5} = 32 / √ 2;
• 12i   (11 ^x – 11) ² = 11 ² + 99;
• 12j   4 ^{x + √ (x^2 – 2)} – 13 * 2 ^{x – 1 + √ (x^2 – 2)} = 12;
• 12k   2 ^{x^2 (x^2 + 1)} = 64 * 5 ^{x^4 + x^2 – 6};
• 12l   3 (x + 1) (3 x + 2) ^x – (x – 1) ^x = (3 x + 2) ^{x + 1};
• 12m   2 sin ² x + 5 sin x cos x + 3 cos ² x = 0;
• 12n   11 sin ² x + 0.5 sin (2 x) - 6 cos ² x = - 4;