Math Antics: The Distributive Property in Algebra

Introduction

• The Distributive Property is a useful tool in Algebra.
• Allows distributing a factor to each member of a group being added or subtracted.
• Works with both known numbers and unknown variables.

Basics of the Distributive Property

• Expression: 3(x + 6)
  • Can't simplify the group since x is unknown.
  • Use the Distributive Property to eliminate the group: 3x + 18.
• Algebraic expression: a(b + c) = ab + ac
  • Factor a is multiplied by the entire group and each member individually.
• Applies to addition and subtraction but not multiplication or division within the group.

Examples

• Example 1: 2(x + y + z) becomes 2x + 2y + 2z
• Example 2: 10(a - b + 4) becomes 10a - 10b + 40
• Example 3: a(x - y + 2) becomes ax - ay + a*2*

Terms in Polynomials

• Group members added or subtracted are terms in polynomials.
• Example: 2(3x + 5y)
  • Distribute 2 gives: 6x + 10y

More Complex Examples

• Example 1: 4(x^2 + 3x - 5)
  • Distribute 4: 4x^2 + 12x - 20
• Example 2: x(x^2 - 8x + 2)
  • Distribute x: x^3 - 8x^2 + 2x

Reverse Distributing (Factoring Out)

• Example: 4x^3 + 4x^2 + 4x
  • Factor out 4: 4(x^3 + x^2 + x)
• Common Factor Example: 8x + 6y + 4z
  • Common factor 2: 2(4x + 3y + 2z)
• Variables Common Factor: ax^2 + ax + a
  • Factor out a: a(x^2 + x + 1)

Conclusion

• Understanding the Distributive Property is fundamental for manipulating algebraic expressions.
• Practice with problems to reinforce understanding.

For more learning, visit mathantics.com