Wayground
Fundamental Theorem of Algebra - Printable Mathematics Worksheets - Wayground

Fundamental Theorem of Algebra

11 questions

10th - 10th Grade

Mathematics

Fundamental Theorem of Algebra is a Grade 10 mathematics worksheet with 11 multiple-choice questions on polynomial roots and zeros. Students identify complete lists of possible rational zeros, use a polynomial’s degree to determine the maximum number of roots, and interpret zeros shown on graphs. The worksheet also asks students to recognize repeated zeros from a graph’s bounce at the x-axis and connect graph features to a polynomial in factored form. These varied questions help students link algebraic reasoning with visual interpretation of polynomial functions. It is a free printable worksheet with a complete answer key, suitable for guided practice, independent review, or a quick check of understanding.

Worksheet settings

Font size

S
M
L
XL
Worksheets

Fundamental Theorem of Algebra

Total questions: 11

Name
Class
Date
1.
Which of the following is a COMPLETE  list of all possible Rational Zeros?
f(x) = 2x3 + 5x2 - 9x + 5
a)
±1, ±5, ±1/2, ±5/2
b)
±1, ±2, ±5, ±1/2, ±5/2
c)
±1, ±5
d)
±1, ±2, ±5, ±1/2, ±5/2, ±2/5
2.
Which of the following is a COMPLETE  list of all possible Rational Zeros?
f(x) = 3x3 + 2x2 - 6x + 7
a)
±1, ±7, ±1/3, ±7/3
b)
±1, 7
c)
±1, ±3, ±7, ±7/3
d)
±1, ±3, ±7, ±1/3, ±7/3
3.

List the possible rational zeros of

f(x) = x3 - 4x2 -4x + 16

a)

1, 2, 4, 8, 16

b)

±1,±2,±4,±8,±16\pm1,\pm2,\pm4,\pm8,\pm16

c)

±1,±4,±14\pm1,\pm4,\pm\frac{1}{4}

d)

-2 and-4

4.

The degree of the polynomial determines the max number of roots.

a)

True

b)

False

5.

According to the Fundamental Theorem of Algebra,

how many roots does g(x) = 23x2 - 24x3 + 3x -9 have?

a)

2

b)

23

c)

3

d)

24

6.
What are the zeros of the function?
a)
4,1
b)
3,1,-2,-5
c)
-3, -1, 2, 5
d)
-2,4
7.

How many zeroes are shown in the graph?

a)

1

b)

2

c)

3

d)

4

8.

If a zero is present twice, how will this affect the graph of the function?

a)

The graph cuts through the x-axis

b)

The graph touches and turns on the x-axis (bounces off)

c)

The graph won't get near the x-axis at that point.

9.

Based on the shape of the graph, which root is there twice?

a)

-1

b)

2

c)

4

d)

-4

10.
What are the zeros of the graph?
a)

-4, -1, 2, 1, 1

b)
-4, -1, 1, 2
c)

4, 1, -2, -1, -1

d)
4, 1, -1, -2
11.
What is the equation in factored form of this graph?
a)
f(x) = (x-4)(x-1)2(x+2)(x+4)
b)
f(x) = (x-4)(x+1)2(x+2)(x+4)
c)
f(x) = (x-4)(x-1)2(x-2)(x+4)
d)
f(x) = (x+4)(x+1)2(x-2)(x-4)
Answer Key

Fundamental Theorem of Algebra

Total questions: 11

1.
a) ±1, ±5, ±1/2, ±5/2
2.
a) ±1, ±7, ±1/3, ±7/3
3.
b)

±1,±2,±4,±8,±16\pm1,\pm2,\pm4,\pm8,\pm16

4.
a)

True

5.
c)

3

6.
c) -3, -1, 2, 5
7.
d)

4

8.
b)

The graph touches and turns on the x-axis (bounces off)

9.
a)

-1

10.
a)

-4, -1, 2, 1, 1

11.
d) f(x) = (x+4)(x+1)2(x-2)(x-4)

Browse by subjects

Browse by grades

Browse by format

FAQs

What does the Fundamental Theorem of Algebra worksheet cover?

This Grade 10 worksheet uses 11 multiple-choice questions to practise identifying possible rational zeros, determining how a polynomial’s degree relates to its number of roots, and reading zeros from graphs. Students also interpret repeated roots and use graph zeros and multiplicity to select the matching factored-form equation.

How can I use this worksheet to teach rational zeros and graph multiplicity?

Begin by modeling how the leading coefficient and constant term generate a complete list of possible rational zeros, including both positive and negative values. Then use the worksheet’s graph questions to discuss x-intercepts, repeated zeros, and why a graph that touches and turns at the x-axis indicates multiplicity before students select a factored form.

What mistakes should I watch for when students complete this worksheet?

Students may omit negative or fractional candidates when listing possible rational zeros, or confuse the polynomial’s degree with a coefficient when counting roots. In the graph items, watch for confusing zeros with other graph features, missing a repeated zero, and choosing the wrong signs when converting zeros into factors such as (x - r).

How do I assign and grade this Fundamental Theorem of Algebra worksheet?

Wayground provides this worksheet as a printable PDF and as a digital quiz, and it includes a complete answer key for all 11 multiple-choice questions. You can grade printable submissions by scanning student work with the Wayground for Teachers app, while the digital version supports online completion and grading.

How does this worksheet connect a graph’s zeros to a polynomial in factored form?

The later multiple-choice questions ask students to read zeros from a graph, identify which root occurs twice from the graph’s shape, and choose the corresponding factored equation. This sequence reinforces that each x-intercept contributes a factor and that a repeated zero appears with a repeated factor.

Where can I find more worksheets like this on Wayground?

Wayground offers a broad collection of free printable worksheets and practice problems across subjects, with downloadable PDFs and answer keys to support essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

© 2026 Wayground Inc.

Accessibility and Inclusion