Student assessment
Short pre-test, full diagnostic, and student score profile.
Download student PDFPolynomial Skills Diagnostic Test, Scoring Guide, and Free PDF
Use a 10-question pre-test or a fuller 25-question polynomial diagnostic, score each skill area separately, and assign the exact remediation students need instead of restarting the entire unit.
In short: The total score tells you whether a student is broadly ready. The section scores tell you what to teach next. A student can score reasonably well overall and still need a focused assignment in factoring, division, or graph behavior. The most common single cause of a low total is incomplete factoring, so start with the factoring polynomials guide if that section comes back red.
On this page
Download the Polynomial Diagnostic Pack
All files are formatted for US letter paper. The editable Word version lets teachers replace numbers, remove topics, or create a retest without rebuilding the layout.
Teacher answer key
Answers, skill coding, thresholds, and follow-up checks.
Download answer key PDFEditable version
Printer-friendly Word file that can be adapted for a class.
Download editable DOCXComplete bundle
Student pages, graph remediation, and teacher key in one PDF.
Download complete bundleGraph matching
A separate activity for zeros, multiplicity, and end behavior.
Download graph activity PDFWhy this works better than a generic unit test
A unit test usually produces one number. This diagnostic produces five decisions. It separates foundational notation and evaluation from operations, factoring, division, and graph interpretation, so a weak synthetic-division score does not erase evidence that the student can multiply and factor well.
The questions are deliberately mixed in difficulty but narrow in purpose. Teachers can use the short form before instruction, then use the longer form after review or before an exam.
Every question and answer on this page was checked for accuracy before publishing, and the four graphs below are plotted directly from their equations, so each marked zero sits exactly on the curve.
Short Polynomial Pre-Test: 10 Questions
Use this form when time is tight. Score each skill pair separately. A total below 7, or a score of 0 in any pair, is a good reason to use the full diagnostic.
- 1Foundations
Rewrite 4 - 3x³ + 2x - x² in standard form. Then state the degree and leading coefficient.
- 2Foundations
Let p(x) = 2x² - 5x + 1. Find p(-2).
- 3Operations
Simplify (3x² - 4x + 7) + (2x² + x - 5).
- 4Operations
Multiply (x - 3)(x + 5).
- 5Factoring
Factor 6x³ - 24x completely.
- 6Factoring
Factor 2x² + 7x + 3.
- 7Division
Divide x³ + 2x² - 5x - 6 by x + 3.
- 8Division
Find the remainder when p(x) = x³ - 4x + 1 is divided by x - 2.
- 9Graphs
List the zeros and multiplicities of f(x) = (x + 1)²(x - 4).
- 10Graphs
For g(x) = -2(x + 2)(x - 1)², describe the end behavior and what the graph does at each zero.
Full Polynomial Unit Diagnostic: 25 Questions
The longer version gives enough evidence to make a targeted assignment. Allow 35-45 minutes and ask students to show work, especially on subtraction, factoring, and division.
- 1Foundations
Rewrite 7 - 2x⁴ + 5x² - x in standard form. State the degree, leading coefficient, and constant term.
- 2Foundations
Classify 3x⁵ - x² + 4 by degree and number of terms.
- 3Foundations
Circle the expression that is not a polynomial: 4x³ - 2x + 1, 3/x + x², 5, or 2x⁴ - 7x².
- 4Foundations
For -6x³ + 2x² - 9, state the coefficient of x², the coefficient of x, and the constant term.
- 5Foundations
Let p(x) = 3x² + x - 4. Find p(-2).
- 6Foundations
Let q(x) = 4x² - 8x + 3. Find q(1/2).
- 7Foundations
If f(x) = x² - 3x + 2 and g(x) = 2x + 1, find f(3) + g(-1).
- 8Operations
Add (4x³ - 2x + 5) + (x³ + 6x² + 3x - 8).
- 9Operations
Subtract (5x² - 3x + 7) - (2x² + 4x - 1).
- 10Operations
Two regions have areas 3x² - 2x + 6 and x² + 5x - 4. Write a polynomial for their total area.
- 11Operations
Multiply 3x(2x² - x + 4).
- 12Operations
Multiply (x + 6)(x - 2).
- 13Operations
Multiply (2x - 3)(x² + x + 4).
- 14Operations
Expand (3x - 5)².
- 15Factoring
Factor 12x⁴ - 18x³.
- 16Factoring
Factor x² - x - 20.
- 17Factoring
Factor 6x² + 13x + 6.
- 18Factoring
Factor 4x³ - 36x completely.
- 19Factoring
Factor x³ + 2x² - 9x - 18 completely.
- 20Division
Divide 2x³ + 3x² - 11x - 6 by x + 3.
- 21Division
Divide x³ - 5x + 7 by x - 2. Write the quotient and remainder.
- 22Division
Is x - 4 a factor of p(x) = x³ - 6x² + 5x + 12? Show the quickest check.
- 23Graphs
For f(x) = (x + 3)(x - 1)², list the zeros and state whether the graph crosses or touches at each one.
- 24Graphs
For g(x) = -x³ + 4x, find the zeros and describe the end behavior.
- 25Graphs
Choose the equation whose graph has a zero at -2 where it touches, a zero at 1 where it crosses, and left-down/right-up end behavior: A. (x + 2)²(x - 1), B. -(x + 2)(x - 1)², C. (x + 2)(x - 1), D. -(x + 2)²(x - 1)².
Skills Breakdown and Assignment System
Enter the number correct in each section. The calculator reports an overall band and flags the smallest useful next assignment. The bands below and the calculator use the same cut scores.
| Skill area | Points | Secure | Developing | Needs work | Assign this resource |
|---|---|---|---|---|---|
| Foundations Questions 1-7 | 7 | 6-7 | 4-5 | 0-3 | Standard form and evaluation refresher |
| Operations Questions 8-14 | 7 | 6-7 | 4-5 | 0-3 | Polynomial operations guide |
| Factoring Questions 15-19 | 5 | 4-5 | 3 | 0-2 | Factoring flowchart and worksheets |
| Division Questions 20-22 | 3 | 3 | 2 | 0-1 | Polynomial division guide |
| Zeros and graphs Questions 23-25 | 3 | 3 | 2 | 0-1 | Graph-matching activity |
Teacher score calculator
Enter the five section scores.
Nothing is uploaded or stored. The calculator runs in the browser.
Targeted Remediation Chart
Each card matches one needs-work band in the table above. Assign one card, not the whole unit.
Standard Form and Evaluation Refresher
Assign when students miss ordering terms, missing coefficients, or substitution with negative numbers.
Write every missing power with a zero coefficient.
Order from greatest exponent to least.
Use parentheses around negative inputs before evaluating.
Polynomial Operations Guide
Assign when adding, subtracting, or multiplying breaks down.
Line up like powers before combining.
Distribute a subtraction sign to every term.
After multiplying, combine only matching powers.
Factoring Flowchart and Worksheets
Assign when students stop after one step or reach for a pattern before pulling out a common factor.
Take out the greatest common factor first.
Match what is left to a pattern: difference of squares, trinomial, or grouping.
Multiply back to confirm nothing is left to factor.
Polynomial Division Guide
Assign when students omit placeholders, confuse the divisor's sign, or do not interpret a remainder.
Insert zero coefficients for missing powers.
For x - a, use a in synthetic division.
Check: divisor × quotient + remainder.
Graph-Matching Activity
Match each graph to one equation from the bank, then justify the match using multiplicity and end behavior. This is the recommended assignment for a score of 0-1 in the zeros-and-graphs section. The graphs are unlabeled on purpose, so students have to read the behavior at each zero rather than the equation above the picture.
Equation bank
- y = -(x + 2)(x - 1)2
- y = (x + 2)(x - 1)
- y = -(x + 2)2(x - 1)2
- y = (x + 2)2(x - 1)
Graph A
Marked zeros: x = -2 (left) and x = 1 (right). Grid squares are one unit.
Graph B
Marked zeros: x = -2 (left) and x = 1 (right). Grid squares are one unit.
Graph C
Marked zeros: x = -2 (left) and x = 1 (right). Grid squares are one unit.
Graph D
Marked zeros: x = -2 (left) and x = 1 (right). Grid squares are one unit.
What students write for each graph
1. The zeros. 2. Crosses or touches at each zero, and the multiplicity that explains it. 3. End behavior on the left and the right. 4. The sign of the leading coefficient and the parity of the degree.
Teacher key: graph matching
Graph A → equation 4: y = (x + 2)²(x - 1)
Touches at -2 (multiplicity 2), crosses at 1 (multiplicity 1), degree 3 with a positive leading coefficient, so the graph falls on the left and rises on the right.
Graph B → equation 1: y = -(x + 2)(x - 1)²
Crosses at -2, touches at 1, degree 3 with a negative leading coefficient, so the graph rises on the left and falls on the right.
Graph C → equation 2: y = (x + 2)(x - 1)
Degree 2 with a positive leading coefficient. Both zeros have multiplicity 1, so the graph crosses at both and both ends rise.
Graph D → equation 3: y = -(x + 2)²(x - 1)²
Degree 4 with a negative leading coefficient. Both zeros have multiplicity 2, so the graph touches at both and both ends fall.
Teacher Answer Key
The printable teacher key includes the same answers in a cleaner scoring format. The sections below are collapsed so students do not see answers immediately, and the "Print student pages" button at the top of this page leaves them out of the printout.
Short pre-test answers
Questions 1-10
1. -3x³ - x² + 2x + 4; degree 3; leading coefficient -3.
Foundations: Arrange terms from greatest exponent to least.
2. 19.
Foundations: 2(-2)² - 5(-2) + 1 = 8 + 10 + 1.
3. 5x² - 3x + 2.
Operations: Combine like terms.
4. x² + 2x - 15.
Operations: Distribute each term.
5. 6x(x - 2)(x + 2).
Factoring: Take out 6x, then factor the difference of squares.
6. (2x + 1)(x + 3).
Factoring: The middle terms 6x and x add to 7x.
7. x² - x - 2.
Division: Synthetic division with -3 gives remainder 0.
8. 1.
Division: Use p(2): 8 - 8 + 1 = 1.
9. x = -1, multiplicity 2; x = 4, multiplicity 1.
Graphs: The exponent on each factor gives its multiplicity.
10. Left end up, right end down; crosses at x = -2; touches and turns at x = 1.
Graphs: Odd degree with a negative leading coefficient; odd multiplicity crosses and even multiplicity touches.
Full diagnostic answers
Questions 1-25
1. -2x⁴ + 5x² - x + 7; degree 4; leading coefficient -2; constant 7.
Foundations: Order terms by descending exponent.
2. Fifth-degree trinomial.
Foundations: The greatest exponent is 5 and there are three nonzero terms.
3. 3/x + x².
Foundations: The term 3/x is 3x⁻¹, which has a negative exponent.
4. 2, 0, and -9.
Foundations: A missing x-term has coefficient 0.
5. 6.
Foundations: 3(4) - 2 - 4 = 6.
6. 0.
Foundations: 4(1/4) - 8(1/2) + 3 = 1 - 4 + 3.
7. 1.
Foundations: f(3) = 2 and g(-1) = -1.
8. 5x³ + 6x² + x - 3.
Operations: Combine terms with matching exponents.
9. 3x² - 7x + 8.
Operations: Distribute the subtraction sign through the second polynomial.
10. 4x² + 3x + 2.
Operations: Add the two area expressions.
11. 6x³ - 3x² + 12x.
Operations: Distribute 3x to every term.
12. x² + 4x - 12.
Operations: The middle terms are -2x and 6x.
13. 2x³ - x² + 5x - 12.
Operations: Distribute both terms, then combine like terms.
14. 9x² - 30x + 25.
Operations: Use (a - b)² = a² - 2ab + b².
15. 6x³(2x - 3).
Factoring: The greatest common factor is 6x³.
16. (x - 5)(x + 4).
Factoring: The numbers -5 and 4 multiply to -20 and add to -1.
17. (3x + 2)(2x + 3).
Factoring: Split 13x as 9x + 4x, then group.
18. 4x(x - 3)(x + 3).
Factoring: Take out 4x, then factor x² - 9.
19. (x + 2)(x - 3)(x + 3).
Factoring: Group, then factor the remaining difference of squares.
20. 2x² - 3x - 2.
Division: Synthetic division with -3 gives remainder 0.
21. x² + 2x - 1 with remainder 5.
Division: Include the missing 0x² term before dividing.
22. Yes. p(4) = 0.
Division: By the Remainder Theorem, remainder 0 means x - 4 is a factor.
23. x = -3, crosses; x = 1, touches and turns.
Graphs: Multiplicity 1 is odd; multiplicity 2 is even.
24. Zeros: -2, 0, 2. Left end up and right end down.
Graphs: Factor -x(x - 2)(x + 2); the degree is odd and the leading coefficient is negative.
25. A. (x + 2)²(x - 1).
Graphs: Positive cubic end behavior; even multiplicity at -2 and odd multiplicity at 1.
How to Use the Diagnostic
Before a unit
Give the 10-question pre-test. Begin the unit normally when the student earns 9-10. At 7-8, assign one warm-up from the weakest pair. At 0-6, use the full diagnostic before planning review.
Before an exam or new course
Give the 25-question form. Start with any red section, not with the first chapter in the textbook. Recheck that skill with two fresh questions after remediation.
What usually goes wrong, in order
Four failure points account for most of the lost points on this diagnostic: dropping the sign when distributing a subtraction, stopping after one factoring step instead of factoring completely, omitting placeholder zeros in division, and treating every zero as a crossing. Each remediation card above is written for one of those four, which is why a red section usually needs a 20-minute fix rather than a reteach of the whole unit.
Standards, Sources, and How This Was Checked
The assessment samples polynomial arithmetic, the relationship between zeros and factors, the Remainder Theorem, polynomial division, and graph behavior. Those topics align most directly with CCSS HSA.APR.A.1, B.2-3, and D.6 and HSF.IF.C.7.c.
All 35 assessment questions and answers were checked for accuracy before publishing and prepared from previous class notes and assigned homework. The four activity graphs are plotted directly from their equations at a fixed scale, and each marked zero is placed at the root, so the picture and the algebra agree. The page uses the Schema.org Quiz type to describe the learning resource. Published August 2, 2026; last reviewed August 2, 2026.
Classroom Use and Link Credit
Teachers may print, edit, and hand out the files for classroom or home use. Please link to this guide instead of reposting the PDFs on another website. That keeps the answer key, corrections, and updated versions in one place.
Suggested credit:
<a href="https://burketutoringinfremont.com/polynomial-diagnostic-test/">Polynomial Skills Diagnostic Test</a> by Berke Sahbazoglu, Burke Tutoring in FremontRelated Worksheets and Guides
These internal links are deliberately limited to the next likely step after a polynomial diagnostic.
Common Questions
Should every student take both versions?
No. Use the short form to screen. Use the longer form when the short score is below 7, one skill pair is 0/2, or you need a more defensible assignment plan.
Can the total score be used as a grade?
It is better used as a planning tool. The section pattern matters more than one percentage, and some sections contain fewer questions than others.
Can students use calculators?
The default version is designed without a calculator. A teacher may allow one when computation is not the skill being measured.
How soon should I retest?
Use two or three fresh questions after the targeted assignment. A full repeat is usually unnecessary unless several sections were weak.
About the Author
Berke Sahbazoglu
Berke has taught K-12 math and science for more than 10 years, logging over 6,000 tutoring hours with 200+ students. He holds a BA in Biochemistry from Washington University in St. Louis and an MS in Bioinformatics from UMGC, and he writes and checks every worksheet published on this site.
This diagnostic is organized around the moments that most often stall a polynomial unit: sign errors during subtraction, incomplete factoring, missing placeholders in division, and the jump from factors to graph behavior.
Page Updates
- August 2, 2026 - published the 10-question pre-test, 25-question diagnostic, teacher answer key, editable version, score calculator, and remediation chart.
- August 2, 2026 - added the graph-matching activity, classroom reuse language, standards alignment, and structured data.
- August 2, 2026 - replotted the four activity graphs from their equations so every marked zero lands on the curve, moved all labels out of the plot area into a key beneath each figure, separated the equation bank from the graphs so the activity is no longer self-answering, aligned the score calculator with the published cut scores, added a student-only print option, and corrected the service-area links.
- August 2, 2026 - fixed the shaded header carrying over into the skill-area column of the score table, and rebuilt the factoring card so all four remediation cards share the same layout.
- August 2, 2026 - replaced the author initials with Berke's photo from his tutor bio page.
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