New York State Regents Examinations Algebra II (Regents Examination) Syllabus

Algebra II (Regents Examination) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Algebra II (Regents Examination) in New York State Regents Examinations, not a summary of it.

1. 1

   Polynomial and Rational Functions

   3 topics

   • Polynomial Arithmetic and Factoring
     • Operations on polynomials and synthetic division
     • Factoring by grouping and sum/difference of cubes
     • Remainder and factor theorems
   • Graphs of Polynomial Functions
     • Zeros, multiplicity, and end behavior
   • Rational Expressions and Equations
     • Simplifying and operating on rational expressions
     • Solving rational equations and extraneous roots
2. 2

   Radicals, Complex Numbers, and Equations

   3 topics

   • Rational Exponents and Radicals
     • Converting between radical and exponent form
     • Solving radical equations
   • Complex Number System
     • Operations with imaginary and complex numbers
     • Complex solutions of quadratics
   • Solving Polynomial Equations
     • Finding real and complex roots of higher-degree equations
3. 3

   Exponential and Logarithmic Functions

   3 topics

   • Exponential Functions and Modeling
     • Continuous growth and the number e
     • Compound interest applications
   • Logarithms and Their Properties
     • Logarithm definition and change of base
     • Product, quotient, and power laws
   • Solving Exponential and Logarithmic Equations
     • Using logarithms to solve for exponents
4. 4

   Trigonometric Functions

   3 topics

   • The Unit Circle and Radian Measure
     • Reference angles and exact values
     • Converting between degrees and radians
   • Graphing Trigonometric Functions
     • Amplitude, period, frequency, and phase shift
     • Modeling periodic phenomena
   • Trigonometric Identities
     • Pythagorean identity and applications
5. 5

   Functions, Sequences, and Series

   3 topics

   • Transformations and Inverse Functions
     • Shifts, stretches, and reflections of parent functions
     • Finding and verifying inverses
   • Sequences and Series
     • Arithmetic and geometric series and sums
     • Sigma notation
   • Comparing and Building Functions
     • Modeling with combinations of function types
6. 6

   Statistics, Probability, and Inference

   4 topics

   • Normal Distribution
     • z-scores and the empirical rule
     • Areas under the normal curve
   • Study Design and Sampling
     • Surveys, experiments, and observational studies
     • Randomization and bias
   • Simulation and Inference
     • Margin of error and confidence in results
     • Evaluating treatment significance via simulation
   • Conditional Probability
     • Independence and the multiplication rule

Algebra II (Regents Examination) flashcards for New York State Regents Examinations

23 of 57 cards from the Algebra II (Regents Examination) deck — real questions with worked answers.

1. State the Remainder Theorem for a polynomial p(x) divided by (x - a).

   When p(x) is divided by (x - a), the remainder equals p(a).

2. State the Factor Theorem.

   (x - a) is a factor of polynomial p(x) if and only if p(a) = 0.

3. Factor the difference of two cubes: a^3 - b^3.

   a^3 - b^3 = (a - b)(a^2 + ab + b^2).

4. Factor the sum of two cubes: a^3 + b^3.

   a^3 + b^3 = (a + b)(a^2 - ab + b^2).

5. For a polynomial in standard form, how do the degree and leading coefficient determine end behavior?

   Even degree: both ends go the same way (up if leading coefficient > 0, down if < 0). Odd degree: ends go opposite ways (up-right/down-left if coefficient > 0, reversed if < 0).

6. What does the multiplicity of a real zero tell you about the graph at that x-intercept?

   Odd multiplicity: the graph crosses the x-axis. Even multiplicity: the graph touches and turns around (is tangent) at that intercept.

7. What is the maximum number of turning points (relative extrema) of a polynomial of degree n?

   At most n - 1 turning points.

8. How many roots (counting multiplicity and complex roots) does a degree-n polynomial have, per the Fundamental Theorem of Algebra?

   Exactly n roots in the complex number system, counting multiplicities.

9. What values must be excluded from the domain of a rational expression?

   Any value of the variable that makes the denominator equal to zero.

10. What is the first step when adding or subtracting rational expressions with unlike denominators?

    Find the least common denominator (LCD), then rewrite each expression with that common denominator before combining numerators.

11. When solving a rational equation, why must you check solutions, and what are rejected solutions called?

    Multiplying out can introduce values that make a denominator zero; those invalid answers are called extraneous solutions and must be rejected.

12. Rewrite the radical expression as a rational exponent: the n-th root of a^m.

    a^(m/n).

13. Simplify (using rational-exponent rules): a^(1/2) · a^(1/3).

    a^(1/2 + 1/3) = a^(5/6).

14. What does a negative rational exponent mean, e.g. a^(-m/n)?

    a^(-m/n) = 1 / a^(m/n), the reciprocal of the positive-exponent expression.

15. How do you rationalize a denominator containing a single square root, such as 1/√a?

    Multiply numerator and denominator by √a to get √a / a.

16. Define the imaginary unit i and give the values of i^2, i^3, and i^4.

    i = √(-1); i^2 = -1; i^3 = -i; i^4 = 1.

17. What is the complex conjugate of a + bi, and what is the product (a + bi)(a - bi)?

    The conjugate is a - bi; their product is a^2 + b^2 (a real number).

18. How do you divide complex numbers, e.g. simplify (a + bi)/(c + di)?

    Multiply numerator and denominator by the conjugate of the denominator (c - di), then simplify to standard a + bi form.

19. State the quadratic formula and what the discriminant b^2 - 4ac reveals.

    x = (-b ± √(b^2 - 4ac)) / (2a). Discriminant > 0: two real roots; = 0: one repeated real root; < 0: two complex conjugate roots.

20. What does the Rational Root Theorem state about possible rational zeros of a polynomial with integer coefficients?

    Any rational zero has the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

21. If a polynomial with real coefficients has the root 2 + 3i, what other root must it have?

    Its complex conjugate, 2 - 3i (complex roots occur in conjugate pairs).

22. Give the general form of an exponential function and state the condition on the base.

    f(x) = a·b^x, where a ≠ 0 and the base b > 0, b ≠ 1.

23. In y = a·b^x, how do you tell exponential growth from decay?

    Growth when b > 1; decay when 0 < b < 1.

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