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Rational Root Theorem Test Answer Key with Scoring – Precalculus

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

Precalculus Math Test Answer Keys – Mathbot, the Math Made Easy robot coach, with free printable test answer keys for Precalculus
Looking for a rational root theorem test answer key? Then this answer key gives the solutions and the scoring guide of the Precalculus test on Polynomial and Rational Functions: Zeros and Asymptotes. Each solution shows the work with a scoring note. Award method points separately from final-answer points, so a small arithmetic slip does not erase credit for a correct sign chart or division.

Standards: A-APR.B.2, A-APR.B.3, F-IF.C.7c, F-IF.C.7d, N-CN.C.9 · Level: Grade 12 Precalculus

Question 1 Find All Zeros (2 points)

Candidates ±1, ±2, ±3, ±6, ±1/2, ±3/2. Since P(2) = 16 + 4 − 26 + 6 = 0, divide to get 2x2 + 5x − 3 = (2x − 1)(x + 3).

Zeros: 2, 1/2, −3.

Scoring: 1 point for a valid first zero with work; 1 point for the remaining two zeros.

Question 2 Conjugate Zeros and Intercepts (2 points)

  1. 1 − 3i and 2i.
  2. All four zeros are non-real, so the graph has 0 x-intercepts.

Scoring: 1 point per part.

Question 3 Zeros and End Behavior from a Graph (2 points)

−6−5−4−3−2−112−30−20−10102030touchcross
The touch point has even multiplicity.
  1. −4 has multiplicity 2 (touch) and 1 has multiplicity 1 (cross).
  2. Odd degree, positive leading coefficient: falls on the left, rises on the right. P(x) = (x + 4)2(x − 1) (check: P(0) = −16).

Scoring: 1 point per part.

Question 4 Slant and Vertical Asymptotes (2 points)

Divide: 4x2 − 5x + 3 = (4x − 1)(x − 1) + 2. The remainder is 2, so f(x) = 4x − 1 + 2x − 1.

Slant asymptote y = 4x − 1; vertical asymptote x = 1.

Scoring: 1 point for each asymptote.

Question 5 Polynomial Inequality (2 points)

Factor: x(2x + 5)(x − 3) ≤ 0. Critical numbers −5/2, 0, 3. Signs from left to right: −, +, −, +.

Solution: (−∞, −5/2] ∪ [0, 3].

Scoring: 1 point for the factorization and critical numbers; 1 point for the correct intervals.

Question 6 Rational Inequality (3 points)

Subtract 3: x + 4 − 3(x − 1)x − 1 = 7 − 2xx − 1 < 0.

Critical numbers 1 and 7/2. Signs: − for x < 1, + on (1, 7/2), − for x > 7/2. Both endpoints are excluded.

(−∞, 1) ∪ (7/2, ∞).

Scoring: 1 point for combining into one fraction; 1 point for critical numbers and signs; 1 point for the interval answer.

Question 7 Build a Polynomial (2 points)

The conjugate −3i is also a zero, giving the factor x2 + 9. Then P(x) = (x + 1)2(x2 + 9) = (x2 + 2x + 1)(x2 + 9).

x4 + 2x3 + 10x2 + 18x + 9.

Scoring: 1 point for the conjugate factor; 1 point for the correct expansion.

Question 8 Analyze a Rational Function (3 points)

−6−4−22468−10−5510x = 1y = x + 2
Asymptotes and intercepts.
  1. The denominator is 0 at x = 1.
  2. x2 + x − 6 = (x + 2)(x − 1) − 4, so the quotient is x + 2: y = x + 2.
  3. Set the numerator to 0: (x + 3)(x − 2) = 0, so −3 and 2.

Scoring: 1 point per part.

Question 9 Unit Cost Over Time (2 points)

  1. A(n) = 5 + 1200/n, so the asymptote is y = 5: with very many items the average cost approaches $5.
  2. 5 + 1200/n ≤ 15 gives 1200 ≤ 10n, so n ≥ 120 items.

Scoring: 1 point per part.

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