Real Numbers and Decimal Expansions

Real Numbers

• Include all rational and irrational numbers
• Can be represented on the number line
• Examples: 1, 2, 3, π, e, √2

Irrational Numbers

• Cannot be expressed as a finite decimal or fraction
• Have an infinite number of digits that never repeat
• Examples: π, e, √2

Decimal Expansions

• A way to represent real numbers in base 10
• Can be terminating (e.g. 1/2 = 0.5) or non-terminating (e.g. 1/3 = 0.333...)
• Non-terminating decimals can be repeating (e.g. 1/3) or non-repeating (e.g. π)

Rational Numbers

• Can be expressed as a finite decimal or fraction (a/b, where a and b are integers)
• Examples: 1, 2, 3, 1/2, 3/4
• Can be represented as equivalent ratios

Laws of Exponents

• Product of Powers: a^m * a^n = a^(m+n)
• Power of a Product: (a*b)^m = a^m * b^m
• Power of a Power: (a^m)^n = a^(m*n)
• Zero Exponent: a^0 = 1 (where a is a non-zero number)
• Negative Exponent: a^(-m) = 1/a^m

Real Numbers

• Include all rational and irrational numbers
• Can be represented on the number line
• Examples: 1, 2, 3, π, e, √2

Irrational Numbers

• Cannot be expressed as a finite decimal or fraction
• Have an infinite number of digits that never repeat
• Examples: π, e, √2

Decimal Expansions

• A way to represent real numbers in base 10
• Can be terminating (e.g. 1/2 = 0.5) or non-terminating (e.g. 1/3 = 0.333...)
• Non-terminating decimals can be repeating (e.g. 1/3) or non-repeating (e.g. π)

Rational Numbers

• Can be expressed as a finite decimal or fraction (a/b, where a and b are integers)
• Examples: 1, 2, 3, 1/2, 3/4
• Can be represented as equivalent ratios

Laws of Exponents

Product of Powers

• a^m * a^n = a^(m+n)

Power of a Product

• (a*b)^m = a^m * b^m

Power of a Power

• (a^m)^n = a^(m*n)

Zero Exponent

• a^0 = 1 (where a is a non-zero number)

Negative Exponent

• a^(-m) = 1/a^m

CH Polynomials: Key Concepts and Formulas

Cubic Polynomials

• A cubic polynomial has a degree of three, meaning the highest power of the variable is three.
• The general form of a cubic polynomial is: ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
• Examples of cubic polynomials include x^3 + 2x^2 - 7x + 1 and 2x^3 - 3x^2 - x + 4.

Factorization

• Factorization is the process of expressing a polynomial as a product of simpler polynomials.
• A polynomial can be factorized if it can be expressed as a product of binomials or trinomials.
• The example x^2 + 5x + 6 can be factorized as (x + 3)(x + 2).

Remainder Theorem

• The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is f(a).
• The theorem allows for evaluating a polynomial at a given value of x without actually dividing the polynomial.

Linear Algebra

• CH polynomials can be represented as vectors in a vector space.
• The set of all CH polynomials forms a vector space, with operations such as addition and scalar multiplication.
• CH polynomials can be represented as matrices, and operations such as multiplication can be performed using matrix multiplication.

Graphing Polynomials

• The graph of a polynomial is a continuous curve that opens upward or downward.
• The x-intercepts of the graph represent the roots of the polynomial.
• The y-intercept represents the constant term of the polynomial.

Factor Theorem

• The Factor Theorem states that if (x - a) is a factor of a polynomial f(x), then f(a) = 0.
• The theorem can be used to find the roots of a polynomial by finding the values of x that make the polynomial equal to zero.

Finding the Value of a and b

• In a quadratic polynomial ax^2 + bx + c, the values of a and b can be found using the formulas:
  • a = (sum of the roots) / 2
  • b = product of the roots
• These formulas can be used to find the values of a and b in a CH polynomial.