Simplifying Algebraic Expressions

Key Concepts

• Combining Like Terms: To simplify expressions, combine terms with the same variable.
• Distributive Property: A(b + c) = Ab + Ac.
• Monomials: Expressions with one term.
• Binomials: Expressions with two terms.
• Trinomials: Expressions with three terms.
• Exponents: When multiplying like bases, add exponents; when dividing, subtract exponents.

Example 1: Simplifying an Expression

• Given: 5x + 8 - 2x + 5
  • Combine like terms:
    • 5x - 2x = 3x
    • 8 + 5 = 13
  • Final answer: 3x + 13

Example 2: More Complex Expression

• Given: 3x + 5y - 9x + 7y
  • Combine like terms:
    • 3x - 9x = -6x
    • 5y + 7y = 12y
  • Final answer: -6x + 12y (also can be written as 12y - 6x)

Example 3: Distributing

• Given: 9(5x + 4)
  • Distribute 9:
    • 9 * 5x = 45x
    • 9 * 4 = 36
  • Final answer: 45x + 36

Example 4: Distributing with Variables

• Given: 7(3x^2 - 8x + 2)
  • Distribute 7:
    • 7 * 3x^2 = 21x^2
    • 7 * -8x = -56x
    • 7 * 2 = 14
  • Final answer: 21x^2 - 56x + 14*

Example 5: Combining and Distributing

• Given: 5(3x + 4) - 7x + 8
  • Distribute:
    • 5 * 3x = 15x
    • 5 * 4 = 20
  • Combine like terms:
    • 15x - 7x = 8x
    • 20 + 8 = 28
  • Final answer: 8x + 28

Example 6: More Complex Distribution

• Given: 5(2x^3 + 5x^2 - 8) - 3(4x^2 + 5x + 9)
  • Distribute both:
    • First: 10x^3 + 25x^2 - 40
    • Second: -12x^2 - 15x - 27
  • Combine like terms:
    • 10x^3 + (25 - 12)x^2 - 15x - (40 + 27)
  • Final answer: 10x^3 + 13x^2 - 15x - 67

Multiplying Monomials

• When multiplying variables, add the exponents:
  • Example: x^2 * x^3 = x^{2+3} = x^5*

Dividing Monomials

• When dividing variables, subtract the exponents:
  • Example: x^8 / x^3 = x^{8-3} = x^5

Example 7: Fractions and Terms

• Given: 36x^7 / 4x^3
  • Divide constants: 36 / 4 = 9
  • Subtract exponents: 7 - 3 = 4
  • Final answer: 9x^4

Example 8: Multiple Terms Division

• Given: (36x^3 + 18x^2 + 15x) / 3x
  • Separate into fractions:
    • 36x^3 / 3x + 18x^2 / 3x + 15x / 3x
  • Simplify each:
    • 12x^2 + 6x + 5
  • Final answer: 12x^2 + 6x + 5

FOIL Method for Binomials

• To multiply two binomials:
  • First, Outer, Inner, Last terms.
  • Example: (3x + 5)(2x - 4):
    • 3x*2x + 3x*(-4) + 5*2x + 5*(-4)
    • Combine like terms.

Conclusion

• Simplifying algebraic expressions involves combining like terms, using the distributive property, and applying rules for exponents during multiplication and division. Practice with various examples to reinforce these concepts.