Algebra 2 Zeros and Graphs of Polynomial Functions Test Solutions and Scoring (PDF)

Algebra 2 math test solutions: Zeros and Graphs of Polynomial Functions – Maths PDF

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This polynomial zeros test answers for Algebra 2 covers zeros and graphs of polynomial functions (Common Core A-APR.B.3, F-IF.C.7c), with a free PDF to print.

Use this answer key and scoring guide to grade the test on zeros and graphs of polynomial functions (A-APR.B.3). Each solution shows the full reasoning, and the scoring note explains how many points to award for correct setup, correct work, and a correct final answer. Partial credit rewards a sound method even when a small arithmetic slip changes the result.

Algebra 2 · Common Core State Standards: A-APR.B.3, F-IF.C.7c

Polynomial zeros test answers: worked solutions

Question 1 – Zeros from factors [2 pts]

\(x = 0\) (multiplicity 1), \(x = -\frac{3}{2}\) (multiplicity 1), \(x = 7\) (multiplicity 2).

\(0, -\frac{3}{2}, 7\)

Scoring: 1 point for the three zeros, 1 point for the multiplicities.

Question 2 – Factor and solve [3 pts]

\(x^{2}(x – 4) – 9(x – 4) = (x – 4)(x^{2} – 9) = (x – 4)(x – 3)(x + 3)\).

\(x = 4, 3, -3\)

Scoring: 1 point for grouping, 1 point for the factored form, 1 point for the zeros.

Question 3 – Multiplicity on a graph [2 pts]

Crossing means odd multiplicity (1 each) and touching means even multiplicity (2 at \(x = 2\)). Total degree 4.

Zeros -3, 1, 2.

\(f(x) = \frac{1}{4}(x + 3)(x – 1)(x – 2)^{2}\)

Scoring: 1 point for the multiplicities, 1 point for the factored form.

Question 4 – End behavior [2 pts]

Odd degree, positive leading coefficient: falls to the left, rises to the right.
Even degree, negative leading coefficient: falls on both ends.

(a) down left, up right; (b) down on both ends

Scoring: 1 point for each part.

Question 5 – Rational Root Theorem [3 pts]

Candidates: \(\pm 1, \pm 2, \pm 4, \pm 8\). \(p(1) = 1 + 1 – 10 + 8 = 0\).

Divide by \(x – 1\): \(x^{2} + 2x – 8 = (x + 4)(x – 2)\).

\(x = 1, 2, -4\)

Scoring: 1 point for the candidates, 1 point for finding a zero and dividing, 1 point for all three zeros.

Question 6 – Polynomial from a graph point [2 pts]

\(f(x) = a(x + 2)(x – 1)(x – 3)\). At \(x = 0\): \(a(2)(-1)(-3) = 6a = -12\), so \(a = -2\).

\(f(x) = -2(x + 2)(x – 1)(x – 3)\)

Scoring: 1 point for the setup with a, 1 point for a = -2 and the equation.

Question 7 – Sign chart [2 pts]

Completed table.

\(f(-2) = (-1)(-4)(-6) = -24\); \(f(0) = 8\); \(f(3) = -4\); \(f(5) = 18\).

-24, 8, -4, 18

Scoring: 1 point for the four values, 1 point for the positions.

Question 8 – Box volume [2 pts]

Zeros: \(x = 0, 8, 5\). The cut must be less than half the shorter side, 5 inches.

Zeros 0, 5, 8; domain 0 < x < 5

Scoring: 1 point for the zeros, 1 point for the domain.

Question 9 – Complex zero [2 pts]

The conjugate \(1 – 2i\) is also a zero.

\((x – 4)(x – (1 + 2i))(x – (1 – 2i))\). The quadratic part is \(x^{2} – 2x + 5\).

\(1 – 2i; f(x) = (x – 4)(x^{2} – 2x + 5)\)

Scoring: 1 point for the conjugate, 1 point for the polynomial.

Total score

Total

20 points

Key facts behind this polynomial zeros test answers

Key facts

Zeros of p(x) are x-intercepts; x – r is a factor when p(r) = 0.
Odd multiplicity: the graph crosses; even multiplicity: it touches and turns.
End behavior follows the leading term a·x^n.
Rational Root Theorem: candidates are ± (factors of constant) / (factors of leading coefficient).
A degree n polynomial has n complex zeros counting multiplicity.

How to use this polynomial zeros test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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