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SAT Math: Polynomials (Drill 2)

Drill 2 · Math · Polynomials

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About This Drill

SAT Math: Polynomials (Drill 2) is a Math practice drill covering Polynomials. It contains 5 original questions developed by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

SAT polynomial questions cover adding, subtracting, and multiplying polynomials, factoring into linear and quadratic factors, identifying real zeros, applying the Remainder Theorem, and analyzing end behavior based on the degree and leading coefficient.

Questions & Explanations

Question 1. If p(x) = x³ − 6x² + 11x − 6, which of the following is a factor of p(x)?

  • A) (x − 4)
  • B) (x + 1)
  • C) (x − 2) ✓
  • D) (x + 6)

Explanation: By the Factor Theorem, (x − a) is a factor if p(a) = 0. Test x = 2: p(2) = 8 − 24 + 22 − 6 = 0. So (x − 2) is a factor. The full factorization is (x − 1)(x − 2)(x − 3).

Question 2. What is the remainder when p(x) = x⁴ − 3x² + 5 is divided by (x + 1)?

  • A) 3 ✓
  • B) 5
  • C) 7
  • D) −1

Explanation: By the Remainder Theorem, the remainder is p(−1) = (−1)⁴ − 3(−1)² + 5 = 1 − 3 + 5 = 3.

Question 3. Which of the following is equivalent to (2x² + 3x − 1) − (x² − 4x + 5)?

  • A) x² − x + 4
  • B) x² + 7x − 6 ✓
  • C) 3x² − x − 6
  • D) x² + 7x + 4

Explanation: Distribute the negative sign: 2x² + 3x − 1 − x² + 4x − 5. Combine like terms: (2x² − x²) + (3x + 4x) + (−1 − 5) = x² + 7x − 6.

Question 4. The polynomial g(x) = 2x³ − x² + 4x − 7 has a leading term of 2x³. Which of the following describes the end behavior of g(x)?

  • A) As x → −∞, g(x) → +∞ and as x → +∞, g(x) → +∞
  • B) As x → −∞, g(x) → −∞ and as x → +∞, g(x) → −∞
  • C) As x → −∞, g(x) → −∞ and as x → +∞, g(x) → +∞ ✓
  • D) As x → −∞, g(x) → +∞ and as x → +∞, g(x) → −∞

Explanation: The leading term is 2x³. Odd degree with a positive leading coefficient means: as x → −∞, g(x) → −∞ and as x → +∞, g(x) → +∞. Choices A and B describe even-degree behavior.

Question 5. The polynomial p(x) has zeros at x = −4, x = 1, and x = 3. If the leading coefficient is 1, which of the following is p(x)?

  • A) x³ − 13x + 12 ✓
  • B) x³ − 7x + 12
  • C) x³ + 13x − 12
  • D) x³ − 12

Explanation: p(x) = (x + 4)(x − 1)(x − 3). Multiply (x − 1)(x − 3) = x² − 4x + 3. Then (x + 4)(x² − 4x + 3) = x³ − 4x² + 3x + 4x² − 16x + 12 = x³ − 13x + 12.