Algebra II (US High School)
US High School Algebra 2 covering polynomials, rational and radical functions, exponentials and logarithms, trigonometry, complex numbers, matrices, conic sections, sequences and series, and statistics with probability — aligned with Common Core (CCSS.MATH.CONTENT.HSA/HSF/HSS).
Ämne: Matematik · Nivå: Gymnasium (16–19) · 449 kort
Innehåll
- A polynomial in one variable is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where the coefficients are real numbers and n is a non-negative integer.
- The degree of a polynomial is the highest exponent of the variable. For example, 4x⁵ − 2x³ + 7 has degree 5.
- The leading coefficient of a polynomial is the coefficient of the term with the highest degree.
- A polynomial is in standard form when its terms are written in descending order of degree.
- The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has at least one complex root.
- A polynomial of degree n has exactly n complex roots, counted with multiplicity.
- The Remainder Theorem: when polynomial p(x) is divided by (x − a), the remainder is p(a).
- The Factor Theorem: (x − a) is a factor of polynomial p(x) if and only if p(a) = 0.
- The Rational Root Theorem: if a polynomial with integer coefficients has a rational root p/q in lowest terms, then p divides the constant term and q divides the leading coefficient.
- Synthetic division is a shortcut for dividing a polynomial by a linear factor (x − a), using only the coefficients.
- The end behavior of a polynomial is determined by its leading term: the leading coefficient's sign and whether the degree is even or odd.
- A polynomial of degree n has at most n − 1 turning points (relative maxima or minima) on its graph.
- Complex conjugate roots theorem: if a polynomial with real coefficients has a complex root a + bi, then a − bi is also a root.
- The sum of cubes factorization: a³ + b³ = (a + b)(a² − ab + b²).
- The difference of cubes factorization: a³ − b³ = (a − b)(a² + ab + b²).
- The Binomial Theorem expands (a + b)ⁿ using binomial coefficients C(n, k) from Pascal's Triangle.
- Pascal's Triangle: each entry is the sum of the two entries directly above it. The n-th row gives the coefficients of (a + b)ⁿ.
- The multiplicity of a root r of polynomial p(x) is the number of times (x − r) appears as a factor.
- At a root of even multiplicity, the graph touches the x-axis but does not cross it. At a root of odd multiplicity, the graph crosses the x-axis.
- The imaginary unit i is defined by i² = −1, equivalently i = √(−1).
- A complex number is written in standard form as a + bi, where a is the real part and b is the imaginary part.
- The complex conjugate of a + bi is a − bi. Multiplying a complex number by its conjugate gives a real result: (a + bi)(a − bi) = a² + b².
- The modulus (absolute value) of a complex number a + bi is |a + bi| = √(a² + b²).
- Powers of i cycle in a period of 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1.
- Complex numbers are added componentwise: (a + bi) + (c + di) = (a + c) + (b + d)i.
- Complex multiplication uses FOIL with i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
- To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator to rationalize.
- In the complex plane (Argand diagram), the horizontal axis is the real axis and the vertical axis is the imaginary axis. The point a + bi corresponds to the coordinates (a, b).
- The discriminant b² − 4ac of a quadratic ax² + bx + c determines the nature of its roots: positive (two real), zero (one repeated real), negative (two complex conjugates).
- A rational function is a ratio of two polynomials: f(x) = p(x)/q(x), where q(x) is not the zero polynomial.
- The domain of a rational function excludes all values of x for which the denominator equals zero.
- A vertical asymptote of a rational function occurs at x = a if the denominator is zero at a and the numerator is not zero there (no common factor canceled).
- A hole (removable discontinuity) occurs at x = a when (x − a) is a common factor of numerator and denominator and cancels.
- Horizontal asymptotes of a rational function depend on the degrees of numerator (n) and denominator (m): if n < m, y = 0; if n = m, y is the ratio of leading coefficients; if n > m, no horizontal asymptote.
- A slant (oblique) asymptote exists when the degree of the numerator is exactly one more than the degree of the denominator. It is found by polynomial long division.
- To add or subtract rational expressions, find a common denominator first, then combine numerators.
- To multiply rational expressions, multiply numerators and multiply denominators. To divide, multiply by the reciprocal.
- An extraneous solution is a value obtained when solving an equation that does not satisfy the original equation, often introduced by squaring or multiplying by a variable expression.
- A radical function involves a root of a variable expression, such as f(x) = √x or f(x) = ∛(x − 1).
- For a square root function f(x) = √(g(x)), the domain is the set of x for which g(x) ≥ 0.
- Cube root functions have domain and range of all real numbers, because every real number has a unique real cube root.
- Rational exponents: x^(m/n) = (ⁿ√x)^m = ⁿ√(x^m), where n is the index of the radical.
- To rationalize a denominator, multiply numerator and denominator by an expression that eliminates the radical (e.g., by √a for √a, or by the conjugate for a + √b).
- An exponential function has the form f(x) = a·b^x, where a ≠ 0, b > 0, and b ≠ 1.
- If b > 1 the function b^x is increasing and represents exponential growth. If 0 < b < 1 the function is decreasing and represents exponential decay.
- The natural base e is an irrational number approximately equal to 2.71828, defined as the limit of (1 + 1/n)ⁿ as n → ∞.
- The natural exponential function f(x) = e^x has the special property that it equals its own derivative.
- The logarithm log_b(x) is the inverse of the exponential b^x: log_b(x) = y means b^y = x.
- The common logarithm log(x) uses base 10. The natural logarithm ln(x) uses base e.
- Product rule for logarithms: log_b(MN) = log_b(M) + log_b(N).