SAT Prep - Passport to Advanced Math (Quadratics, Polynomials, Radicals, Exponential Laws)

SAT – Passport to Advanced Math (Quadratics, Polynomials, Radicals, Exponential Laws)

SAT Passport to Advanced Math Study Guide

Topic: Quadratics, Polynomials, Radicals, Exponential Laws

What This Is

The Passport to Advanced Math section tests your ability to work with complex equations, functions, and expressions—especially quadratics, polynomials, radicals, and exponentials. These concepts appear in ~16 of the 58 Math questions (both calculator and no-calculator sections) and are critical for scoring 600+. Real-world example: Modeling projectile motion (e.g., a ball’s height over time) often uses quadratic equations. A typical SAT question might ask you to rewrite an expression, solve for a variable, or interpret a graph’s key features.

Key Terms & Rules

Quadratics

• Standard Form: ax² + bx + c = 0 (where a ≠ 0). Used for factoring and the quadratic formula.
• Factored Form: a(x – r₁)(x – r₂) = 0 (roots are r₁ and r₂).
• Vertex Form: a(x – h)² + k (vertex at (h, k); axis of symmetry: x = h).
• Quadratic Formula: x = [-b ± √(b² – 4ac)] / (2a) (use when factoring is hard).
• Discriminant: D = b² – 4ac. If D > 0: 2 real roots; D = 0: 1 real root; D < 0: no real roots.
• Sum/Product of Roots: For ax² + bx + c = 0, sum = -b/a, product = c/a.

Polynomials

• Degree: Highest power of x (e.g., x³ + 2x is degree 3).
• End Behavior: For axⁿ:
• n even: Both ends go up (if a > 0) or down (if a < 0).
• n odd: Left end down, right end up (if a > 0) or vice versa.
• Factoring by Grouping: Split the middle term to factor (e.g., x³ + 3x² + 2x + 6 = x²(x + 3) + 2(x + 3) = (x² + 2)(x + 3)).
• Remainder Theorem: If f(x) is divided by (x – a), the remainder is f(a).

Radicals

• Simplifying Radicals: √(ab) = √a × √b (e.g., √12 = √4 × √3 = 2√3).
• Rationalizing Denominators: Multiply numerator/denominator by the radical (e.g., 1/√2 = √2/2).
• Exponent Rules for Radicals: ⁿ√(xᵐ) = x^(m/n) (e.g., √(x³) = x^(3/2)).

Exponentials

• Exponent Rules:
• xᵃ × xᵇ = x^(a+b)
• (xᵃ)ᵇ = x^(ab)
• x⁰ = 1 (for x ≠ 0)
• x^(-a) = 1/xᵃ
• Exponential Growth/Decay: A = P(1 ± r)^t (P = initial amount, r = rate, t = time).
• Logarithm Basics: If aᵇ = c, then logₐ(c) = b. SAT may test log(ab) = log a + log b.

Step-by-Step / Process Flow

Solving a Quadratic Equation (No Calculator)

Question: Solve 2x² – 8x + 6 = 0.

1. Check for factoring: Can you write as (dx + e)(fx + g) = 0?
2. Here: 2x² – 8x + 6 = 2(x² – 4x + 3) = 2(x – 1)(x – 3).
3. Set each factor to zero: x – 1 = 0 → x = 1; x – 3 = 0 → x = 3.
4. Verify with quadratic formula (if stuck):
5. a = 2, b = -8, c = 6.
6. x = [8 ± √(64 – 48)] / 4 = [8 ± √16]/4 = [8 ± 4]/4 → x = 3 or x = 1.

Rewriting Expressions (Calculator Allowed)

Question: Which is equivalent to (3x² + 12x)/(x² + 4x + 4)?

1. Factor numerator and denominator:
2. Numerator: 3x(x + 4).
3. Denominator: (x + 2)² (perfect square).
4. Cancel common factors: (3x(x + 4))/(x + 2)². No common factors → can’t simplify further.
5. Check answer choices: Look for 3x/(x + 2) (common distractor) or (3x² + 12x)/(x + 2)² (correct).

Interpreting Polynomial Graphs

Question: The graph of f(x) = x³ – 2x² – 5x + 6 crosses the x-axis at x = -2, 1, and 3. What is f(4)?

1. Use factored form: f(x) = (x + 2)(x – 1)(x – 3).
2. Plug in x = 4: (4 + 2)(4 – 1)(4 – 3) = 6 × 3 × 1 = 18.

Common Mistakes

1. Mistake: Forgetting the ± in the quadratic formula.

2. Correction: Always write ±√(b² – 4ac). The SAT loves to test if you drop the ±.

3. Mistake: Misapplying exponent rules (e.g., (x²)³ = x⁵).

4. Correction: (x²)³ = x^(2×3) = x⁶. Write out the exponents to avoid errors.

5. Mistake: Canceling terms incorrectly (e.g., (x² + 5x)/(x + 5) = x²).

6. Correction: Factor first: (x(x + 5))/(x + 5) = x (only if x ≠ -5).

7. Mistake: Ignoring domain restrictions (e.g., √(x – 3) requires x ≥ 3).

8. Correction: Always check if the expression is defined for the given x.

9. Mistake: Confusing linear and exponential growth (e.g., y = 2x vs. y = 2ˣ).

10. Correction: Linear grows by +m each step; exponential grows by ×b.

Exam Insights

1. Most-Tested Concepts:
2. Quadratics: Factoring, vertex form, and the quadratic formula (especially with a ≠ 1).
3. Exponentials: Rewriting expressions (e.g., 8ˣ = (2³)ˣ = 2^(3x)) and growth/decay.

4. Radicals: Simplifying and rationalizing denominators (e.g., 1/(3 – √2) → multiply by (3 + √2)).

5. Tricky Distractors:

6. Partial Factoring: The SAT may give 2(x + 3)(x – 2) but ask for 2x² + 2x – 12 (you must expand).

7. Sign Errors: Watch for -b in the quadratic formula or negative exponents (e.g., x^(-2) = 1/x²).

8. Calculator Tips:

9. Use the quadratic formula program (if allowed) to save time.

10. For exponentials, use logarithms (e.g., log(100) = 2 because 10² = 100).

11. Graph Interpretation:

12. The vertex of a parabola is the max/min point.
13. Roots are where the graph crosses the x-axis (set y = 0).

Quick Check Questions

1. Question: If x² – 5x + 6 = 0, what is the sum of the solutions?
2. A) -5
3. B) 5
4. C) 6

5. D) -6
   Answer: B) 5. Sum of roots = -b/a = -(-5)/1 = 5.

6. Question: Which expression is equivalent to (2x⁴y⁻²)/(4x²y³)?

7. A) (x²)/(2y⁵)
8. B) (x²y)/(2)
9. C) (1)/(2x²y⁵)

10. D) (2x²)/(y⁵)
    Answer: A) (x²)/(2y⁵). Subtract exponents: x^(4-2)y^(-2-3) = x²y^(-5); simplify coefficients: 2/4 = 1/2.

11. Question: The function f(x) = 3(2ˣ) models exponential growth. What is f(3)?

12. Answer: 24. f(3) = 3(2³) = 3 × 8 = 24.

Last-Minute Cram Sheet

1. Quadratic Formula: x = [-b ± √(b² – 4ac)] / (2a) ⚠️ Don’t forget the ±!
2. Vertex Form: a(x – h)² + k → vertex at (h, k).
3. Exponent Rules: xᵃ × xᵇ = x^(a+b); (xᵃ)ᵇ = x^(ab).
4. Radicals: √(ab) = √a × √b; rationalize denominators.
5. Sum of Roots: -b/a (for ax² + bx + c = 0).
6. Product of Roots: c/a.
7. Discriminant: D = b² – 4ac → tells you # of real roots.
8. Polynomial End Behavior: Even degree → both ends same; odd degree → ends opposite.
9. Exponential Growth: A = P(1 + r)^t; decay: A = P(1 – r)^t.
10. ⚠️ Common Trap: The SAT loves a ≠ 1 in quadratics—always check!