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Ch 06 - We Distribute, Yet Things Multiply8th Ganita Prakash

Algebraic Identity: Distributive Property of Multiplication

CONCEPT Formula

a(b + c) = ab + ac

States the algebraic identity for distributing a single multiplier over an addition.

Derivation

Geometrically, consider a rectangle of width $a$ and total length $(b+c)$ .
The total area is exactly $a \times (b+c)$ .
The rectangle can be split strictly into two smaller rectangles: one of area $ab$ and another of area $ac$ .
Therefore, $a(b+c) = ab + ac$ .

Component Anatomy:

$a$ : The external multiplier.
$b, c$ : The terms inside the binomial being added.

Boundary Conditions / Domain:

$a, b, c \in \mathbb{R}$
The operation inherently requires a minimum of two terms inside the bracket to distribute over.

Prism Memory Hook

Pass the 'a' to every term inside to stay in the game.

Product Variation: Single Factor Increment by 1

APPLICATION Formula

a(b + 1) = ab + a

Calculates the strict numerical increase of a product $ab$ when the second factor $b$ is increased by exactly 1.

Derivation

Apply the fundamental distributive property to the incremented expression: $a(b + 1)$ .
Distribute $a$ to $b$ yielding the original product: $ab$ .
Distribute $a$ to the increment $1$ yielding: $a$ .
Sum the results to find the expanded state: $ab + a$ .
The product strictly increases by the value of the unaltered factor $a$ .

Component Anatomy:

$ab$ : The initial baseline product.
$a$ : The exact absolute value by which the total product increases.

Boundary Conditions / Domain:

$a, b \in \mathbb{R}$

Prism Memory Hook

A premium mnemonic for this topic is available in the Prism App.

Product Variation: Double Factor Increment by 1

APPLICATION Formula

(a + 1)(b + 1) = ab + a + b + 1

Expands the product algebraically when both constituent factors $a$ and $b$ are strictly increased by 1.

Derivation

Treat the entire first binomial $(a+1)$ as a single rigid term and distribute it over $(b+1)$ .
Yields: $(a+1)b + (a+1)1$ .
Expand further using standard distributivity: $ab + b + a + 1$ .
Rearrange to isolate the original product core: $ab + (a + b + 1)$ .

Component Anatomy:

$ab$ : The original baseline product.
$(a + b + 1)$ : The total absolute increment added to the original product due to the dual shift.

Boundary Conditions / Domain:

$a, b \in \mathbb{R}$

Prism Memory Hook

A premium mnemonic for this topic is available in the Prism App.

Also in this chapter

27 more cards to study in Prism

4Generalized Binomial Expansion: Mixed Variable Increments
5Algebraic Expansion: Binomial Multiplied by Trinomial
6Common Trap: Negative Monomial Distribution
7Polynomial Simplification: Like vs. Unlike Terms Limit
8Standard Identity 1A: Square of a Binomial Sum
9Algebraic Inequality: Square of Sum vs. Sum of Squares
10Standard Identity 1B: Square of a Binomial Difference
11Geometric Proof: Square of a Binomial Difference
12Standard Identity 1C: Product of Sum and Difference
13Algebraic Synthesis: Sum of Conjugate Binomial Squares
14Advanced Generalization: Difference of N-th Powers
15Algorithmic SOP: Rapid Mental Multiplication by 11
16Algebraic Decomposition: Multiplying by Base-10 Offsets
17Algorithmic Squaring: Sridharacharya's Transposition Method
18Heuristic Comparison: Close Products via Midpoints
19System Formulation: Decomposing Integers into Difference of Squares
20Geometric Polynomial Modeling: Border and Path Areas
21Sequence Algebraic Modeling: Geometric Visual Patterns
22Real-World Number Theory: The Calendar Grid Phenomenon
23Algebraic Proof: Validating the 7-Day Calendar Constant
24Generalization Logic: Modeling Consecutive Integers
25Identity Proof: Middle Square vs. Outer Cross-Product
26Parity Proof: Sum of Squares of Consecutive Integers
27Modular Arithmetic: Sum of Remainders Theorem
28Modular Arithmetic: Product of Remainders Theorem
29Algorithmic SOP: Converting Negative Modulo Remainders
30Modulo 8 Theorem: Parity Property of Odd Squares

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8th Ganita Prakash

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