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SAT Math: Polynomials (Drill 1)

Drill 1 · Math · Polynomials

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About This Drill

SAT Math: Polynomials (Drill 1) 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 test addition, subtraction, and multiplication of polynomials, factoring by grouping or special forms, finding zeros, understanding end behavior, and applying the Remainder Theorem to evaluate polynomials without full division.

Questions & Explanations

Question 1. If x = 1 is a zero of p(x) = x³ − 4x² − 7x + 10, what is the sum of all three zeros of p(x)?

  • A) 3
  • B) 7
  • C) 4 ✓
  • D) 10

Explanation: By Vieta's formulas, for a cubic x³ + bx² + cx + d, the sum of the zeros equals −b. Here the sum = −(−4) = 4. The zeros are 1, 5, and −2.

Question 2. When the polynomial p(x) = 2x³ − 5x² + 3x − 7 is divided by (x − 2), what is the remainder?

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

Explanation: By the Remainder Theorem, the remainder is p(2) = 2(8) − 5(4) + 3(2) − 7 = 16 − 20 + 6 − 7 = −5.

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

  • A) x³ + x² − 2x + 12 ✓
  • B) x³ − 2x² + 4x + 12
  • C) x³ + 3x² − 2x + 12
  • D) x³ + x² + 2x + 12

Explanation: Distribute: x(x² − 2x + 4) = x³ − 2x² + 4x. And 3(x² − 2x + 4) = 3x² − 6x + 12. Combine: x³ + (−2x² + 3x²) + (4x − 6x) + 12 = x³ + x² − 2x + 12.

Question 4. The polynomial f(x) = −3x⁴ + 5x³ − x + 2 has a leading term of −3x⁴. Which of the following describes the end behavior of f(x)?

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

Explanation: The leading term is −3x⁴. Even degree with negative coefficient means both ends point downward: as x → ±∞, f(x) → −∞.

Question 5. What are the zeros of p(x) = x³ + 5x² − 9x − 45?

  • A) x = −5, x = −9, x = 1
  • B) x = 5, x = 3, x = −3
  • C) x = −5, x = 3, x = −3 ✓
  • D) x = 5, x = −3, x = −45

Explanation: Factor by grouping: x²(x + 5) − 9(x + 5) = (x² − 9)(x + 5) = (x − 3)(x + 3)(x + 5). The zeros are x = 3, x = −3, and x = −5.