SAT (Scholastic Assessment 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 SAT (Scholastic Assessment Test), not a summary of it.

1. 1

   Equivalent Expressions

   3 topics

   • Polynomial operations
     • Adding, subtracting, and multiplying polynomials
     • Factoring techniques
   • Rational expressions
     • Simplifying and combining
     • Division and remainders
   • Exponent rules and radicals
     • Integer and rational exponents
     • Simplifying radical expressions
2. 2

   Quadratic Equations and Functions

   4 topics

   • Solving quadratics
     • Factoring
     • Completing the square
     • Quadratic formula
   • The discriminant and number of solutions
   • Forms of a parabola
     • Vertex form and the vertex
     • Standard and factored forms, intercepts
   • Interpreting quadratic models
     • Maximum and minimum values
3. 3

   Nonlinear Functions

   4 topics

   • Exponential growth and decay
     • Growth and decay factors
     • Comparing linear and exponential models
   • Polynomial and rational functions
     • End behavior and zeros
     • Asymptotes
   • Function notation and transformations
     • Evaluating and composing functions
     • Shifts, stretches, and reflections
   • Nonlinear equations and systems

Math: Advanced Math flashcards for SAT (Scholastic Assessment Test)

23 of 49 cards from the Math: Advanced Math deck — real questions with worked answers.

1. What does it mean to add or subtract polynomials, and what stays the same?

   Combine only like terms (same variable raised to the same power); add or subtract their coefficients. The exponents themselves never change.

2. How do you multiply two binomials like (x + a)(x + b)?

   Use FOIL (First, Outer, Inner, Last): x² + (a + b)x + ab. Multiply each term in the first by each term in the second, then combine like terms.

3. What is the degree of a polynomial, and how do you find it?

   The degree is the highest exponent on the variable in any term. For multivariable terms, it's the largest sum of exponents in a single term.

4. State the difference of squares factoring formula.

   a² − b² = (a + b)(a − b).

5. State the two perfect-square trinomial formulas.

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

6. What is the Remainder Theorem?

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

7. What is the Factor Theorem?

   (x − a) is a factor of polynomial p(x) if and only if p(a) = 0 (i.e., a is a root).

8. When adding rational expressions, what must you do first?

   Find a common denominator (the least common denominator), rewrite each fraction over it, then add the numerators.

9. How do you divide one rational expression by another?

   Multiply by the reciprocal of the divisor (flip the second fraction and multiply), then factor and simplify.

10. What is a key restriction on the domain of a rational expression?

    The denominator can never equal zero; any value making the denominator 0 is excluded from the domain.

11. How do you simplify a rational expression like (x²−9)/(x²−x−6)?

    Factor numerator and denominator, then cancel common factors. Here: (x+3)(x−3)/[(x−3)(x+2)] = (x+3)/(x+2), with x ≠ 3.

12. State the product rule for exponents.

    x^a · x^b = x^(a+b).

13. State the quotient rule for exponents.

    x^a / x^b = x^(a−b).

14. State the power rule for exponents.

    (x^a)^b = x^(ab).

15. What does a negative exponent mean? Give x^(−n).

    A negative exponent means reciprocal: x^(−n) = 1/x^n (for x ≠ 0).

16. What does a fractional exponent mean? Express x^(m/n) as a radical.

    x^(m/n) = the n-th root of x^m = (ⁿ√x)^m. The denominator is the root index, the numerator is the power.

17. What is any nonzero number raised to the power 0?

    x^0 = 1 for any x ≠ 0.

18. How do you simplify √50?

    Factor out perfect squares: √50 = √(25·2) = 5√2.

19. How do you rationalize a denominator like 1/√3?

    Multiply numerator and denominator by √3: 1/√3 = √3/3.

20. State the quadratic formula.

    For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / (2a).

21. What are the main methods for solving a quadratic equation?

    Factoring, using the quadratic formula, completing the square, taking square roots, and graphing to find x-intercepts.

22. What is the Zero Product Property and how is it used to solve quadratics?

    If a·b = 0, then a = 0 or b = 0. After factoring a quadratic to 0, set each factor equal to 0 and solve.

23. How do you solve x² = 49 by taking square roots?

    x = ±√49 = ±7. Remember both the positive and negative roots.

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