Algebraic expression and formulas

Basic Concepts of Algebraic Expressions

  • Variables and Constants

    • A variable is a symbol, such as xx, yy, or zz, representing an unknown number or value.

    • A constant is a fixed value that does not change, such as 55, −3-3, or π\pi.

  • Terms and Coefficients

    • A term is a single number, variable, or product of numbers and variables (e.g., 3x23x^2, −5y-5y, 77).

    • The coefficient is the numerical factor of a variable term. In 4x4x, the coefficient is 44.

  • Types of Algebraic Expressions

    • Monomial: An expression with one term (e.g., 5x35x^3).

    • Binomial: An expression with two terms (e.g., 2x+32x + 3).

    • Trinomial: An expression with three terms (e.g., x2−4x+4x^2 - 4x + 4).

    • Polynomial: An expression with one or more terms containing non-negative integer exponents.

Simplifying Algebraic Expressions

  • Combining Like Terms

    • Like terms have the exact same variables raised to the exact same powers.

    • Example: Simplify 3x+5x−2x3x + 5x - 2x

    • Result: 6x6x

    • Example: Simplify 4x2+2x−x2+74x^2 + 2x - x^2 + 7

    • Result: 3x2+2x+73x^2 + 2x + 7

  • Distributive Property

    • a(b+c)=ab+aca(b + c) = ab + ac

    • Example: Expand 3(2x−5)3(2x - 5)

    • 3(2x−5)=6x−153(2x - 5) = 6x - 15

Key Algebraic Formulas

  • Expansion Formulas

    • Perfect square (addition): (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

    • Perfect square (subtraction): (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

    • Difference of squares: (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2

    • Cube of a sum: (a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

    • Cube of a difference: (a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

  • Factoring Formulas

    • Sum of cubes: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

    • Difference of cubes: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

  • Quadratic Formula

    • For an equation of the form ax2+bx+c=0ax^2 + bx + c = 0:

    • x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}