Simplifying Algebraic Expressions

Overview

• Simplifying algebraic expressions involves combining like terms and distributing when necessary.

Key Concepts

1. Combining Like Terms

• Like terms are terms that contain the same variable.
• Example: 5x and -2x are like terms.
• Constants (numbers without variables) can also be combined.

Example 1

• Expression: 5x + 8 - 2x + 5
  • Combine: 5x and -2x ➔ 3x
  • Combine: 8 and 5 ➔ 13
  • Result: 3x + 13

Example 2

• Expression: 3x + 5y - 9x + 7y
  • Combine: 3x and -9x ➔ -6x
  • Combine: 5y and 7y ➔ 12y
  • Result: -6x + 12y or 12y - 6x

2. Distributive Property

• Used when multiplying a term by an expression in parentheses.
• Formula: a(b + c) = ab + ac

Example 1

• Expression: 9(5x + 4)
  • Distribute: 9 * 5x = 45x, 9 * 4 = 36
  • Result: 45x + 36

3. More Complex Expressions

Example 1

• Expression: 5xy + 6x^2 + 8xy - 9x^2
  • Combine: 5xy and 8xy ➔ 13xy
  • Combine: 6x^2 and -9x^2 ➔ -3x^2
  • Result: 13xy - 3x^2

4. Multiplying Monomials

• When multiplying, add the exponents of like bases.

Example 1

• Expression: x^2 * x^3
  • Result: x^(2+3) = x^5

Example 2

• Expression: 4x^3 * 5x^2
  • Multiply constants: 4 * 5 = 20
  • Add exponents: 3 + 2 = 5
  • Result: 20x^5

5. Dividing Monomials

• Subtract the exponents of like bases.

Example 1

• Expression: x^8 / x^3
  • Result: x^(8-3) = x^5

6. FOIL Method for Binomials

• Used to multiply two binomials.
• Stands for First, Outside, Inside, Last.

Example

• Expression: (3x + 5)(2x - 4)
  • Calculate: 6x^2 (First), -12x (Outside), 10x (Inside), and -20 (Last)
  • Combine like terms: 6x^2 - 2x - 20

7. Multiplying Trinomials

• Multiply each term in one trinomial by each term in the other trinomial.
  • Combine like terms to simplify.

Example

• Expression: (2x^2 - 7x + 4)(3x^2 + 5x + 7)
  • Multiply each term and simplify to get the final expression.

8. Simplifying Rational Expressions

• Divide each term in the numerator by the denominator separately.

Example

• Expression: 36x^3 + 12x / 3x
  • Result: 12x^2 + 4

Practice Problems

1. Simplify: 5 * (3x + 4) - 7x + 8
2. Multiply: (x^2 + 2)(x + 3)
3. Divide: 28x^5y^4 / 7x^2y^2
4. Simplify: 3x^4 - 2x^3 + x^2 / x^2

Conclusion

• Simplifying algebraic expressions requires understanding combining like terms, distributive properties, and applying multiplication and division rules for exponents.