Polynomials and Exponents

• These notes cover polynomials, exponents, simplifying expressions, solving equations, inequalities, and absolute values.

Exponents

• An exponent represents repeated multiplication of a base number.
• a^n indicates that a is multiplied by itself n times, where n is the power and a is the base.
• n can be a negative numbers and/or real numbers.

Rules of Exponents

• Product of Powers: a^x * a^y = a^(x+y) (add exponents when multiplying with the same base).
• Quotient of Powers: a^x / a^y = a^(x-y) (subtract exponents when dividing with the same base)
• Power of a Power: (a^x)^y = a^(x*y) (multiply exponents when raising a power to a power)
• Zero Exponent: a^0 = 1 (any number raised to the power of 0 is 1; 0^0 is undefined)
• Power of a Product: (ab)^x = a^x * b^x (distribute the exponent to each factor in the product)
• Power of a Quotient: (a/b)^x = a^x / b^x (distribute the exponent to both numerator and denominator)
• Negative Exponent: a^(-n) = 1 / a^n (a negative exponent indicates a reciprocal)
• Rational Exponent: (a/b)^n = (a^n) / (b^n)

Polynomials

• A polynomial is an expression with one or more terms, each containing a variable raised to a non-negative integer power and multiplied by a coefficient: an*x^n + an-1*x^(n-1) + ... + a2*x^2 + a1*x + a0
• n is a non-negative integer.
• an is a real number (coefficient).
• Three types of polynomials: monomial, binomial, trinomial.
  • Monomial: a polynomial with one term (e.g., 2x, 3x^2, 7x^5)
  • Binomial: polynomial with two terms (e.g., 2x^2 + x, 5x + 10x^2)
  • Trinomial: a polynomial with three terms (e.g., x^3 + x^2 + 5, x^2 + 2x + 5)
• The degree of a polynomial is determined by its highest power
  • Example: x^4 + 7x^3 + 2x + 8 is of degree 4.

Adding and Subtracting Polynomials

• Combine like terms (terms with the same variable and exponent).
• When subtracting polynomials, distribute the negative sign to all terms in the second polynomial.

Multiplying Polynomials

• Multiply each term of one polynomial by each term of the other polynomial (distributive property)
• Simplify by combining like terms

Special Product Formulas

• Square of a Binomial:
  • (a + b)^2 = a^2 + 2ab + b^2
  • (x + 2)^2 = x^2 + 4x + 4
  • (a - b)^2 = a^2 - 2ab + b^2
  • (x - 3)^2 = x^2 - 6x + 9
• Difference of Squares: (a + b)(a - b) = a^2 - b^2
  • (x + 4)(x - 4) = x^2 - 16

Order of Operations

1. Parentheses ()
2. Brackets []
3. Braces {}
4. Exponents a^n
5. Multiplication and Division (from left to right)
6. Addition and Subtraction (from left to right)

Integral Exponents and Simplification

• Simplify expressions with exponents using the rules of exponents
• Express final answers with positive exponents only.

Solving Equations

• An equation equates two mathematical expressions.
• Follow algebraic manipulation rules to isolate the variable on one side.

Quadratic Equations

• A quadratic equation is in the form ax^2 + bx + c = 0, where a ≠ 0.
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)
• Discriminant: b^2 - 4ac
  • If it's positive: two real solutions.
  • If it's negative: no real solutions.
  • If it's zero: one real solution.
• Solve by Factoring: Factor the quadratic expression and set each factor to zero

Inequalities and Absolute Value

• Inequalities use symbols like >, <, ≥, ≤ to compare expressions
• Interval Notation:
  • Open interval: (a, b) (excludes a and b)
  • Closed interval: [a, b] (includes a and b)
  • Half-open intervals: (a, b] or [a, b)
• When multiplying or dividing an inequality by a negative number, reverse the inequality sign.