Algebra 2 Zeros and Graphs of Polynomial Functions Worksheet (PDF)

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

Cover of the book Algebra 2 Math Worksheets with Answers (PDF)

Download as a PDFThe Algebra 2 worksheets book with answersPrintable PDFSee the book ›

Sommaire

This polynomial zeros worksheet 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.

These 20 problems practice zeros and graphs of polynomial functions (Common Core A-APR.B.3 and F-IF.C.7c). The first problems check factoring to find zeros, the middle problems add multiplicity, end behavior, and the Rational Root Theorem, and the last six are word problems and multi-step tasks that need building polynomials from zeros and modeling a box and a roller coaster.

Work in order and show every step. Write each answer in simplest form, and open the answer key only after you have tried the problem yourself.

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

Polynomial zeros worksheet: exercises 1 to 10

Exercise 1 – Zeros from factored form

Find the zeros of \(f(x) = (x – 3)(x + 5)(2x – 1)\).

Exercise 2 – Factor out an x

Find the zeros of \(x^{3} – 9x\).

Exercise 3 – Trinomial with a common factor

Find the zeros of \(x^{3} + 2x^{2} – 8x\).

Exercise 4 – Factor by grouping

Find all zeros of \(x^{3} + 3x^{2} – 4x – 12\).

Exercise 5 – Quadratic type

Finding zeros by factoring.

Find all real zeros of \(x^{4} – 5x^{2} + 4\).

Exercise 6 – Multiplicity

List the zeros of \(f(x) = (x + 1)^{2}(x – 2)^{3}\) with their multiplicities and say whether the graph crosses or touches the x-axis at each.

Exercise 7 – Read a graph

Graph of a cubic polynomial.

The cubic graphed has leading coefficient 1. Identify each zero and its multiplicity, and write the equation in factored form.

Exercise 8 – End behavior, odd

Describe the end behavior of \(f(x) = -2x^{5} + 3x^{2} – 1\).

Exercise 9 – End behavior, even

End behavior depends on the degree (even or odd) and the sign of the leading coefficient.

Describe the end behavior of \(g(x) = 3x^{4} – x^{3} + 2\) using the panels as a guide.

Exercise 10 – Rational root candidates

List the possible rational zeros of \(2x^{3} – 3x^{2} – 11x + 6\).

Exercises 11 and up

Exercise 11 – Find all zeros, monic

Find all zeros of \(x^{3} – 2x^{2} – 5x + 6\).

Exercise 12 – Find all zeros, non-monic

Testing candidates.

Find all zeros of \(2x^{3} – 3x^{2} – 11x + 6\).

Exercise 13 – Build from zeros

Write a cubic polynomial with leading coefficient 1 and zeros \(2\), \(-1\), and \(4\). Give it in standard form.

Exercise 14 – Double zero and a point

A cubic has a double zero at \(x = 1\), a zero at \(x = -3\), and passes through \((0, 6)\). Write its equation in standard form.

Exercise 15 – Sign chart

Complete the sign chart.

Find the sign of \(f(x) = (x + 1)(x – 2)(x – 4)\) on each interval.

Exercise 16 – Complex zeros

Solve \(x^{3} – 1 = 0\) over the complex numbers.

Exercise 17 – Conjugate zeros

A degree 5 polynomial with real coefficients has zeros \(2\) (multiplicity 2), \(-1\), and \(3i\). Find the remaining zero.

Exercise 18 – Open box

V(x) for the box from an 8 by 12 inch sheet.

A box is made from an 8-inch by 12-inch sheet by cutting a square of side \(x\) from each corner. The volume is \(V(x) = x(12 – 2x)(8 – 2x) = 4x^{3} – 40x^{2} + 96x\). Find the zeros of \(V\) and explain which values of \(x\) make sense.

Exercise 19 – Roller coaster

Height h (in tens of feet) at horizontal distance x (in hundreds of feet).

The height of a roller coaster (in tens of feet) at a horizontal distance of \(x\) hundred feet is \(h(x) = x(x – 4)^{2}\) for \(0 \le x \le 6\). Find where the track is at ground level and how the graph behaves there. Find the height at \(x = 2\).

Exercise 20 – Find the error

Maya says the graph of \(f(x) = (x – 2)^{2}(x + 1)\) crosses the x-axis at \(x = 2\). Decide whether she is right and explain.

Exercise 21 – Odd degree always crosses

Explain why every polynomial of odd degree with real coefficients has at least one real zero.

Key facts behind this polynomial zeros worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the polynomial zeros worksheet as a PDF with the button below and keep it with your notes.

All Algebra 2 chapters: Algebra 2 math.

Voter.. post

Free printable PDF