Multiplying Across a Group
The Distributive Law states that multiplying a number by a sum is the same as multiplying each part of the sum separately.
\[a(b + c) = ab + ac\]
Think of it like a mailman delivering a package to every house in a neighborhood.
Worked Examples
Example 1: Basic Distribution
Expand: \(4(x + 3)\)
- Multiply 4 by \(x\) \(\to 4x\).
- Multiply 4 by 3 \(\to 12\).
- Result: \(4x + 12\)
Example 2: Negative Distribution
Expand: \(-2(3x - 5)\)
- Multiply -2 by \(3x\) \(\to -6x\).
- Multiply -2 by -5 \(\to +10\) (Note the sign change!).
- Result: \(-6x + 10\)
Example 3: Distributing a Variable
Expand: \(x(x + 4)\)
- Multiply \(x\) by \(x\) \(\to x^2\).
- Multiply \(x\) by 4 \(\to 4x\).
- Result: \(x^2 + 4x\)
Example 4: The Invisible One
Expand: \(-(x + 7)\)
- Think of this as \(-1(x + 7)\).
- Distribute the -1: \(-x - 7\).
- Result: \(-x - 7\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often use "Linear Operators" (Lesson 9.1). A defining property of these operators is that they follow the distributive law: \(\hat{A}(\psi_1 + \psi_2) = \hat{A}\psi_1 + \hat{A}\psi_2\). This means that if you want to know how a force affects a "superposition" of two states, you can just calculate its effect on each state separately and add them up. This simple algebraic rule is what makes complex quantum calculations possible.