Algebra 2 Zeros and Graphs of Polynomial Functions Test with Point Values (PDF)

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

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This polynomial zeros test 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.

This 45-minute test on zeros and graphs of polynomial functions has 9 questions worth 20 points in total. The questions move from direct skills to multi-step problems. Show your work, because partial credit is given for a correct setup and sound reasoning.

Time and points

About this test

Grade: Algebra 2
Chapter: Zeros and Graphs of Polynomial Functions
Standards: A-APR.B.3, F-IF.C.7c
Time: 45 minutes
Total: 20 points

Polynomial zeros test: questions

Question 1 – Zeros from factors [2 pts]

Find the zeros of \(g(x) = x(2x + 3)(x – 7)^{2}\) and state each multiplicity.

Question 2 – Factor and solve [3 pts]

Find all zeros of \(x^{3} – 4x^{2} – 9x + 36\) by factoring.

Question 3 – Multiplicity on a graph [2 pts]

Graph of a quartic with leading coefficient 1/4.

The graph of a quartic polynomial crosses the x-axis at \(-3\) and \(1\) and touches it at \(2\). State the multiplicity of each zero and write a possible factored form (leading coefficient \(\frac{1}{4}\)).

Question 4 – End behavior [2 pts]

Describe the end behavior of each function.

\(f(x) = 5x^{3} – 2x\) (1 point)
\(g(x) = -x^{4} + 6x^{2}\) (1 point)

Question 5 – Rational Root Theorem [3 pts]

Use the Rational Root Theorem to find all zeros of \(x^{3} + x^{2} – 10x + 8\).

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

A cubic has zeros \(-2\), \(1\), and \(3\) and passes through \((0, -12)\). Write its equation in factored form.

Question 7 – Sign chart [2 pts]

Evaluate f(x) = (x + 1)(x – 2)(x – 4).

For \(f(x) = (x + 1)(x – 2)(x – 4)\), complete the table and state whether the graph is above or below the x-axis at each x.

Question 8 – Box volume [2 pts]

Open box with height x.

A box is made from a 10-inch by 16-inch sheet by cutting a square of side \(x\) from each corner. The volume is \(V = x(16 – 2x)(10 – 2x)\). Find the zeros of \(V\) and the interval of \(x\) values that make a box.

Question 9 – Complex zero [2 pts]

A cubic polynomial with real coefficients has zeros \(4\) and \(1 + 2i\). Find the third zero and write the polynomial with leading coefficient 1 in factored form.

Key facts behind this polynomial zeros test

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

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All Algebra 2 chapters: Algebra 2 math.

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