Master Module • Curriculum Hardened

Exploring Algebraic Identities

Experience the high-fidelity mathematical architecture of the universe. Every concept, proof, and historical insight in one place.

4.1 The Entry Point

The Magic of Patterns

In this chapter, we transition from linear polynomials to Algebraic Identities—special rules that simplify the most complex calculations.

Example 1: The Square Paradox

Take any 3 consecutive square numbers (e.g., \( 25, 36, 49 \)). Add the smallest and largest, then subtract twice the middle:

\( (25 + 49) - (2 \times 36) = 74 - 72 = \mathbf{2} \)

Wait... the result is always 2! We will soon use algebra to prove why.

4.2 Geometric Proof

Visualising \( (a + b)^2 \)

Imagine a line segment of length \( (a+b) \). A square constructed on this line can be partitioned into four distinct areas.

\( a^2 \)

\( b^2 \)

\( ab \)

\( (a + b)^2 = a^2 + 2ab + b^2 \)

Verification: Does it work for all numbers?

Click to see the rational number check (\( a = -2/3, b = 3/4 \)).

Identity vs Equation

Equation: \( x^2-1=24 \) is true only for \( x = \pm 5 \).
Identity: \( (a+b)^2 \) is true for EVERY value.

Thinking Point

Is \( (a+b)^2 > a^2 + b^2 \)? Yes, if \( a, b \) are positive, because of the extra \( 2ab \) term!

4.3 Strategies

Factorisation & Algebraic Proof

Identities can be reversed to find factors. For example, \( x^2 + 4x + 4 = (x+2)^2 \).

James vs Reshma: Solving \( (a-b)^2(a+b) \)

James expands first, while Reshma groups terms as \( (a-b)[(a-b)(a+b)] = (a-b)(a^2-b^2) \). Both are correct, but Reshma's path is often faster!

Visualising \( (a-b)^2 \)

Subtract the rectangles from a large square of side \( a \):

\( (a-b)^2 = a^2 - ab - b(a-b) = a^2 - 2ab + b^2 \)

4.4 Historical Logic

Trinomials & Śhrīdharāchārya

What if we square three numbers? \( (a+b+c)^2 \)

\( (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \)

Śhrīdharāchārya’s Square Lab (750 CE)

² =

(55+5)(55-5) + 5² = 3025

4.5 & 4.6 Mechanics

Algebra Tiles & Term Splitting

Visualise \( x^2+7x+12 \) as a rectangle with sides \( (x+3) \) and \( (x+4) \).

\( (px + a)(qx + b) = (pq)x^2 + (pb + aq)x + ab \)

Saira's Rectangle Exercise

Saira has \( x^2 + 8x + 15 \). By splitting 8x into 3x and 5x, she finds the dimensions: \( (x+3) \) by \( (x+5) \).

4.7 Volumetric

The Power of Cubes

Expanding into the 3rd dimension: \( (a+b)^3 \) can be split into 2 cubes and 6 cuboids.

\( (a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3 \)

\( x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \)

The Numerical Challenge

If \( x+y+z=10, xyz=25, x^2+y^2+z^2=38 \), then \( x^3+y^3+z^3 = \mathbf{145} \)!

4.8 Rationality

Simplifying Rational Expressions

We simplify rational expressions by cancelling non-zero common factors.

Complex Case:

\(\frac{x^2 - 7x + 12}{5x^2 + 5x - 100} = \frac{(x-3)(x-4)}{5(x-4)(x+5)} = \frac{x-3}{5(x+5)}\)

Chapter Wrap-up

The 11 Master Identities

1. (x + y)² = x² + 2xy + y²
2. (x - y)² = x² - 2xy + y²
3. (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
4. (x + y)(x - y) = x² - y²
5. (x + a)(x + b) = x² + (a + b)x + ab
6. x³ + y³ = (x + y)(x² - xy + y²)
7. x³ - y³ = (x - y)(x² + xy + y²)
8. (x + y)³ = x³ + 3x²y + 3xy² + y³
9. (x - y)³ = x³ - 3x²y + 3xy² - y³
10. x³ + y³ + z³ - 3xyz = (x+y+z)(x²+y²+z²-xy-yz-zx)
11. (ax + b)(cx + d) = acx² + (ad+bc)x + bd

PRACTICE THIS CHAPTER