PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Advanced Math Syllabus

Math: Advanced Math syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Math: Advanced Math in PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test), not a summary of it.

1. 1

   Equivalent Expressions

   3 topics

   • Factoring Polynomials
     • Greatest common factor and grouping
     • Factoring quadratics and special products
   • Operations with Polynomials and Rationals
     • Adding, subtracting, and multiplying polynomials
     • Simplifying rational expressions
   • Exponent Rules and Radicals
     • Integer and rational exponents
     • Simplifying radical expressions
2. 2

   Nonlinear Equations in One Variable

   3 topics

   • Solving Quadratic Equations
     • Factoring, completing the square, and the quadratic formula
     • Using the discriminant to count solutions
   • Radical and Rational Equations
     • Checking for extraneous solutions
   • Exponential Equations
     • Solving with common bases
3. 3

   Nonlinear Functions

   4 topics

   • Quadratic Functions and Parabolas
     • Vertex, axis of symmetry, and zeros
     • Standard, vertex, and factored forms
   • Exponential Functions
     • Growth and decay models
     • Interpreting base and initial value
   • Function Notation and Transformations
     • Evaluating and interpreting f(x)
     • Shifts, stretches, and reflections of graphs
   • Systems Involving a Nonlinear Equation
     • Solving a linear-quadratic system

Math: Advanced Math flashcards for PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test)

23 of 50 cards from the Math: Advanced Math deck — real questions with worked answers.

1. What does it mean to factor a polynomial?

   To write it as a product of simpler polynomials (its factors) so that multiplying them back gives the original polynomial.

2. How do you factor a quadratic of the form x² + bx + c?

   Find two numbers that multiply to c and add to b; then x² + bx + c = (x + p)(x + q) where p·q = c and p + q = b.

3. State the difference of squares factoring formula.

   a² − b² = (a + b)(a − b).

4. State the factoring formulas for a sum and a difference of cubes.

   a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²).

5. State the perfect-square trinomial factoring patterns.

   a² + 2ab + b² = (a + b)²; a² − 2ab + b² = (a − b)².

6. What is the first step you should always try when factoring any polynomial?

   Factor out the greatest common factor (GCF) of all the terms.

7. How do you factor by grouping a four-term polynomial like ax + ay + bx + by?

   Group into pairs, factor each pair's GCF: a(x + y) + b(x + y), then factor out the common binomial: (x + y)(a + b).

8. What does the Factor Theorem state about a root r of a polynomial p(x)?

   If p(r) = 0, then (x − r) is a factor of p(x); conversely, if (x − r) is a factor, then r is a root.

9. How do you add or subtract polynomials?

   Combine like terms (terms with the same variable raised to the same power) by adding or subtracting their coefficients.

10. How do you multiply two binomials (FOIL)?

    Multiply First, Outer, Inner, Last terms, then combine like terms: (a + b)(c + d) = ac + ad + bc + bd.

11. How do you add or subtract two rational expressions?

    Find a common denominator, rewrite each fraction with that denominator, add/subtract the numerators, then simplify.

12. How do you multiply and divide rational expressions?

    To multiply: multiply numerators and multiply denominators, then simplify. To divide: multiply by the reciprocal of the divisor.

13. How do you simplify a rational expression?

    Factor the numerator and denominator completely, then cancel any common factors.

14. State the product rule for exponents.

    xᵃ · xᵇ = xᵃ⁺ᵇ (add exponents when multiplying like bases).

15. State the quotient rule for exponents.

    xᵃ / xᵇ = xᵃ⁻ᵇ (subtract exponents when dividing like bases).

16. State the power rule for exponents.

    (xᵃ)ᵇ = xᵃᵇ (multiply exponents when raising a power to a power).

17. What does a negative exponent mean?

    x⁻ⁿ = 1/xⁿ (a negative exponent gives the reciprocal of the base raised to the positive exponent).

18. What is the value of any nonzero base raised to the zero power?

    x⁰ = 1 (for x ≠ 0).

19. How do you express a radical as a rational (fractional) exponent?

    ⁿ√(xᵐ) = x^(m/n); the denominator is the root index and the numerator is the power.

20. What is the product rule for radicals?

    √a · √b = √(ab), and conversely √(ab) = √a · √b.

21. How do you rationalize a denominator containing a single square root, like 1/√a?

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

22. State the quadratic formula for ax² + bx + c = 0.

    x = (−b ± √(b² − 4ac)) / (2a).

23. What is the discriminant, and what does its sign tell you?

    The discriminant is b² − 4ac. Positive → two real solutions; zero → one real (repeated) solution; negative → no real solutions (two complex).

See more Math: Advanced Math flashcards →

Planning Math: Advanced Math for PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test)

Math: Advanced Math (PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test)) FAQ