VISUALISATIONS OF ALGEBRAIC PROCESSES AND STRUCTURES

VISUAL ALGEBRA I represents a set of systematised algebraic expansions and consists of 54 algebraic visualisations created to give a meaning to:

1a Simple binomial expansions expressed in the following forms:
- a (x + b)
- a (x – b)
- a (- x + b)
- a (- x – b)
and their mirror images which produce identical outcomes:
- - a (- x – b)
- - a (- x + b)
- - a (x – b)
- - a (x + b);
1b Binomial expansions expressed in the following forms:
- a (b x + c)
- a (b x – c)
- a (- b x + c)
- a (- b x – c)
and their opposites which produce identical outcomes:
- - a (- b x – c)
- - a (- b x + c)
- - a (b x – c)
- - a (b x + c);
1c1 Generalised binomial expansions expressed in the following forms:
- p (q + number)
- p (q – number);
1c2 Generalised binomial expansions expressed in the following forms:
- p (q + number × r)
- p (q – number × r);
1c3 Generalised binomial expansions expressed in the following forms:
- x (x + number)
- x (x – number);
1c4 Generalised binomial expansions expressed in the following forms:
- x (x + number × y)
- x (x – number × y);
1c5 Generalised binomial expansions expressed in the following forms:
- x (number₁ × x + number₂)
- x (number₁ × x – number₂);
1c6 Generalised binomial expansions expressed in the following forms:
- number₁ × x (number₂ × x + number₃)
- number₁ × x (number₂ × x – number₃);
1c7 Generalised binomial expansions expressed in the following forms:
- number₁ × x (number₂ × x + number₃ × y)
- number₁ × x (number₂ × x – number₃ × y);
1d1 Expanding and simplifying expressed in the following forms:
- a (x + b) + c;
1d2 Expanding and simplifying (factorable) expressed in the following forms:
- a (x + b) + c x;
1d3 Expanding and simplifying expressed in the following forms:
- a (x + b) + c (x + d);
1d4 Expanding and simplifying (factorable) expressed in the following forms:
- a (x + b) + c (x – d);
1d5 Expanding and simplifying expressed in the following forms:
- a (b x – c) + d (e x + f);
1d6 Expanding and simplifying (factorable) expressed in the following forms:
- a (b x – c) + d (- e x + f);
1d7 Expanding and simplifying expressed in the following forms:
- a (x + b) – c (x + d);
1d8 Expanding and simplifying (factorable) expressed in the following forms:
- a (x – b) – c (x – d);
1d9 Expanding and simplifying expressed in the following forms:
- a (b x + c) – d (e x + f);
1d10 Expanding and simplifying (factorable) expressed in the following forms:
- a (- b x + c) – d (e x – f);
1e Trinomial expansions expressed in the following forms:
- number₁ ( a + number₂ × b + number₃)
- number₁ ( a + number₂ × b – number₃)
- number₁ ( a – number₂ × b – number₃);
1f1 Generalised trinomial expansions expressed in the following forms:
- p ( q + number₁ × r + number₂)
- p ( q + number₁ × r – number₂)
- p ( q – number₁ × r + number₂)
- p ( q – number₁ × r – number₂);
1f2 Generalised trinomial expansions expressed in the following forms:
- x ( x + number₁ × y + number₂)
- x ( x + number₁ × y – number₂)
- x ( x – number₁ × y + number₂)
- x ( x – number₁ × y – number₂);
1f3 Generalised trinomial expansions expressed in the following forms:
- number₁ × x (number₂ × x + number₃ × y + number₄)
- number₁ × x (number₂ × x + number₃ × y – number₄)
- number₁ × x (number₂ × x – number₃ × y + number₄)
- number₁ × x (number₂ × x – number₃ × y – number₄);
2a Two stage expansions expressed in the following forms:
- (a + b) (c + d)
- (a + b) (c – d)
- (a + b) (- c + d)
- (a + b) (- c – d)
- (a – b) (c + d)
- (a – b) (c – d)
- (a – b) (- c + d)
- (a – b) (- c – d)
and their mirror images which produce identical outcomes:
- (- a – b) (- c – d)
- (- a – b) (- c + d)
- (- a – b) (c – d)
- (- a – b) (c + d)
- (- a + b) (- c – d)
- (- a + b) (- c + d)
- (- a + b) (c – d)
- (- a + b) (c + d);
2b1 Quadratic expansions expressed in the following forms:
- (x + a) (x + b)
- (x + a) (x – b)
- (x – a) (x + b)
- (x – a) (x – b);
2b2 Difference of two squares expressed in the following forms:
- (x + a) (x – a)
- (x – a) (x + a);
2b3 Squared sum and squared difference expressed in the following forms:
- (x + a)²
- (x – a)²;
2c Two-step, two-stage expansions expressed in the following forms:
- number × (a + b) (c + d)
- number × (a + b) (c – d)
- number × (a – b) (c + d)
- number × (a – b) (c – d);
2d1 Two-step, two-stage quadratic expansions expressed in the following forms:
- number × (x + a) (x + b)
- number × (x + a) (x - b)
- number × (x – a) (x + b)
- number × (x – a) (x – b);
2d2 Two-step, two-stage quadratic expansions expressed in the following forms:
- number × (x + a) (x – a);
2d3 Two-step, two-stage quadratic expansions expressed in the following forms:
- number × (x + a)²
- number × (x – a)²;
2e1 Examples of quadratic expansions containing:
- (a x + b) (c x + d) and their mirror images (- a x – b) (- c x – d);
2e2 Examples of quadratic expansions containing:
- (a x + b) (c x – d) and their mirror images (- a x – b) (- c x + d);
2e3 Examples of quadratic expansions containing:
- (a x + b) (- c x + d) and their mirror images (- a x – b) (c x – d);
2e4 Examples of quadratic expansions containing:
- (a x + b) (- c x – d) and their mirror images (- a x – b) (c x + d);
2e5 Examples of quadratic expansions containing:
- (a x – b) (c x + d) and their mirror images (- a x + b) (- c x – d);
2e6 Examples of quadratic expansions containing:
- (a x – b) (c x – d) and their mirror images (- a x + b) (- c x + d);
2e7 Examples of quadratic expansions containing:
- (a x – b) (- c x + d) and their mirror images (- a x + b) (c x – d);
2e8 Examples of quadratic expansions containing:
- (a x – b) (- c x – d) and their mirror images (- a x + b) (c x + d);
2e9 Examples of quadratic expansions containing:
- (a x + b)² and their mirror images (- a x – b)²;
2e10 Examples of quadratic expansions containing:
- (a x + b) (a x – b) and their mirror images (- a x – b) (- a x + b);
2e11 Examples of quadratic expansions containing:
- (a x – b)² and their mirror images (- a x + b)²;
2f1 Comparison of differences containing:
- (a + b x) (c x + d)
- a + b x (c x + d) and
- (a + b x) c x + d;
2f2 Comparison of differences containing:
- (a x + b) (c x – d)
- a x + b (c x – d) and
- (a x + b) c x – d;
2f3 Comparison of differences containing:
- (a x – b) (c – d x)
- a x – b (c – d x) and
- (a x – b) c – d x;
3a Multiplication of trinomials expressed in the form:
- (a + b + c) (d + e + f);
3b Squaring trinomials expressed in the form:
- (a + b + c)²;
3c Multiplication of polynomials expressed in the form:
- (a + b + c) (d + e + f + g);
3d Multiplication of polynomials expressed in the form:
- (a + b + c + d) (e + f + g + h);
3e Multiplication of polynomials expressed in the form:
- (a + b + c) (d + e + f + g + h);
3f Squaring polynomials expressed in the form:
- (a + b + c + d)²;
4a Cubic expansions expressed in the form:
- (a + b)³;
4b Cubic expansions expressed in the form:
- (x + 1)³;
4c Cubic expansions expressed in the form:
- (x + 2)³;