SAT Math: Polynomials (Drill 1)

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.