← Back to the Polynomial Functions test
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Each answer lists the reasoning, the final result in bold, and the scoring guide. Give full credit for equivalent forms of a polynomial, and partial credit for a correct method with an arithmetic slip.
Standards: F-IF.C.7c, A-APR.B.3, F-IF.B.4 · Level: Grade 11 Algebra 2
Question 1 Degree and end behavior (2 points)
a) Degree 5, leading coefficient −4.
b) Odd degree with a negative leading coefficient: rises to the left and falls to the right.
Scoring: 1 point each part.
Question 2 Zeros and multiplicity (2 points)
Zero 6: multiplicity 2, touches. Zero −1: multiplicity 1, crosses. Zero –92: multiplicity 3, crosses (flattened).
6 (2, touches); −1 (1, crosses); –92 (3, crosses)
Scoring: 1 point for the zeros, 1 point for multiplicities and behavior.
Question 3 Factor to find zeros (2 points)
x(x2 − 7x + 10) = x(x − 2)(x − 5).
0, 2, 5
Scoring: 1 point for factoring, 1 point for the zeros.
Question 4 Build a polynomial (3 points)
f(x) = a(x + 4)(x − 1)(x − 3). f(0) = a(4)(−1)(−3) = 12a = −24, so a = −2.
(x + 4)(x − 1)(x − 3) = x3 − 13x + 12.
f(x) = −2x3 + 26x − 24
Scoring: 1 point for the factored form, 1 for a, 1 for the expansion.
Question 5 Read a graph (3 points)
a) Zeros −3, 1, 4. The graph touches at 1, so 1 has even multiplicity.
b) Zeros −3, 1 (twice), 4 give degree 4; both ends rise, so the leading coefficient is positive.
c) p(x) = a(x + 3)(x − 1)2(x − 4) with a > 0.
Scoring: 1 point per part.
Question 6 Sign analysis (3 points)
Zeros: −5, 2 (even), 6. Test points: f(−6) = (−1)(64)(−12) > 0; f(0) = (5)(4)(−6) < 0; f(4) = (9)(4)(−2) < 0; f(7) = (12)(25)(1) > 0.
f(x) > 0 for x < −5 and x > 6
Scoring: 1 point for the zeros and multiplicities, 1 for the sign tests, 1 for the answer.
Question 7 Modeling an open box (3 points)
a) V(x) = x(9 − 2x)(15 − 2x), domain 0 < x < 4.5.
b) V(x) = 4x3 − 48x2 + 135x.
c) V(1) = 1(7)(13) = 91; V(2) = 2(5)(11) = 110; V(3) = 3(3)(9) = 81. x = 2 gives the greatest volume, 110 cubic inches.
Scoring: 1 point per part.
Question 8 Reasoning about graphs (2 points)
a) No. The ends of an odd-degree graph point in opposite directions, so the graph must cross the x-axis at least once.
b) Yes. A degree 5 polynomial has at most 4 turning points, and x5 − 5x3 + 4x has exactly 4.
Scoring: 1 point each, including the explanation.