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Remainder Theorem and Factor Theorem - Printable Mathematics Worksheets - Wayground

Remainder Theorem and Factor Theorem

20 questions

10th - 11th Grade

Mathematics

Remainder Theorem and Factor Theorem is a Grade 10–11 mathematics worksheet that develops students’ ability to interpret polynomial remainders and connect them to factors and solutions. Students use results from polynomial and synthetic division to decide whether a number is a solution, whether a linear expression is a factor, and what a nonzero remainder reveals. The 20-question set includes 16 multiple-choice questions, 3 matching items, and 1 multiple-select question. Matching tasks reinforce the relationship between values such as 5 or -7 and factors such as x - 5 or x - 7, while other questions require students to complete or interpret synthetic division and identify the remainder. This free printable worksheet includes a complete answer key and is useful for guided practice, independent work, or checking students’ understanding of the two theorems.

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Worksheets

Remainder Theorem and Factor Theorem

Total questions: 20

Name
Class
Date
1.

According to this work, is x-3 a factor of p(x)=5x3−2x2+x−120p\left(x\right)=5x^3-2x^2+x-120  ?

a)

Yes, x-3 is a factor of p(x) since the remainder is 0.

b)

No, x-3 is not a factor of p(x) since the remainder is 0.

2.

This work shows x2−3x+5÷x−2x^2-3x+5\div x-2  . What information does this reveal?

a)

x-2 is not a factor of x2−3x+5x^2-3x+5   since the remainder is 3.

b)

x-2 is a factor of x2−3x+5x^2-3x+5   since the remainder is 3.

3.

This work shows x2−3x+5÷x−2x^2-3x+5\div x-2  . What information does this reveal?

a)

2 is a solution of x2−3x+5x^2-3x+5   since the remainder is 3.

b)

2 is not a solution of x2−3x+5x^2-3x+5   since the remainder is 3.

4.

Match the following appropriately.

It is important you can recognize the connection between factors and solutions.

1 is a solution of p(x).

2x - 3 is a factor of p(x).

3 is a solution of p(x).

x-1 is a factor of p(x).

3/2 is a solution of p(x).

x - 3 is a factor of p(x).

0 is a solution of p(x).

x is a factor of p(x).

5.

Match the following appropriately.

It is important you can recognize the connection between factors and solutions.

x + 3 is a factor of p(x).

-3 is a solution of p(x).

x - 5 is a factor of p(x).

5 is a solution of p(x).

3x - 5 is a factor of p(x).

5/3 is a solution of p(x).

x + 5 is a factor of p(x).

-5 is a solution of p(x).

6.

What do we know about p(x)= 3x4+2x3−5x+43x^4+2x^3-5x+4  based off of the work shown here?

Read carefully. Only one statement is true.

a)

1 is a solution of p(x)

b)

1 is not a solution of p(x)

c)

x -1 is a factor of p(x)

d)

x+1 is not a factor of p(x)

7.
Question Image

Match the following:

is a factor of 5x2−10x−405x^2-10x-40  

x + 2...

is a solution of 5x2−10x−405x^2-10x-40  

-2...

0

The remainder is...

x+2

The work provided shows the polynomial 5x2 - 10x - 40 being divided by...

8.

This work shows x3+2x2−58x−35÷x−7x^3+2x^2-58x-35\div x-7  . What does this reveal?

More than one answer is correct.

a)

x-7 is a factor of p(x).

b)

x+7 is a factor of p(x).

c)

-7 is a solution of p(x).

d)

7 is a solution of p(x).

9.

Which statement accurately describes what this work demonstrates?

a)

x+6 is a factor of x3−x2−40x +13x^3-x^2-40x\ +13  .

b)

x+6 is not a factor of x3−x2−40x +13x^3-x^2-40x\ +13  .

c)

x-6 is not a factor of x3−x2−40x +13x^3-x^2-40x\ +13  .

d)

x-6 is a factor of x3−x2−40x +13x^3-x^2-40x\ +13  .

10.

Complete the problem shown here.

Use the remainder to determine whether 5 is a solution of p(x)=2x3−12x2+11x−5p\left(x\right)=2x^3-12x^2+11x-5  .

a)

5 is a solution of p(x) since the remainder is 0.

b)

5 is a solution of p(x) since the remainder is 100.

c)

5 is not a solution of p(x) since the remainder is 0.

d)

5 is not a solution of p(x) since the remainder is 100.

11.

Complete the problem shown here.

Use the remainder to determine whether x-5 is a solution of p(x)=2x3−12x2+11x−5p\left(x\right)=2x^3-12x^2+11x-5  .

a)

x-5 is a factor of p(x) since 5 was a solution.

b)

x-5 is not a factor of p(x) since 5 was not a solution.

c)

x+5 is not a factor of p(x) since 5 was not a solution.

d)

x+5 is a factor of p(x) since 5 was a solution.

12.

The work here shows that dividing x2+16x^2+16  by x+4 results in a remainder of 32. Which statement is true based upon that work shown here?

a)

-4 is a solution of x2+16x^2+16  

b)

4 is a solution of x2+16x^2+16

c)

x+4 is not a factor of x2+16x^2+16  

d)

x-4 is not a factor of x2+16x^2+16  

13.

This work shows 4x2+17÷x+24x^2+17\div x+2  .

a)

True

b)

False

14.

Complete the synthetic division and use the remainder to determine a true statement concerning p(x) and its solutions.

a)

-2 is a solution of p(x) since the remainder is equal to 0.

b)

-2 is not a solution of p(x) since the remainder is equal to 1.

c)

-2 is not a solution of p(x) since the remainder is equal to 33.

d)

2 is not a solution of p(x) since the remainder is equal to -1.

15.

Complete the rest of this synthetic division problem.

Determine the remainder.

a)

The remainder is 18.

b)

The remainder is 0.

c)

The remainder is 108.

d)

The remainder is 28.

16.

Based on the remainder you found, which statement is true?

a)

3 is a solution of 2x3−15x+92x^3-15x+9  .

b)

3 is not a solution of 2x3−15x+92x^3-15x+9  .

c)

-3 is a solution of 2x3−15x+92x^3-15x+9  .

d)

-3 is not a solution of 2x3−15x+92x^3-15x+9  .

17.

Based on the remainder you found, which statement is true?

a)

x-3 is not a factor of 2x3−15x+92x^3-15x+9  .

b)

x-3 is a factor of 2x3−15x+92x^3-15x+9  .

c)

x+3 is not a factor of 2x3−15x+92x^3-15x+9  .

d)

x+3 is a factor of 2x3−15x+92x^3-15x+9  .

18.

Assume that a polynomial has a solution at the x-value 4.

Which of the following must be a factor of that same polynomial?

a)

4x

b)

x-4

c)

x+4

19.

Assume that x-2 is a factor of a polynomial. Which x-value must be a solution of that same polynomial?

a)

The x-value -2

b)

The x-value 2

c)

The x-value 0

20.

Finish dividing p(x)=5x3+34x2+26x+Cp\left(x\right)=5x^3+34x^2+26x+C   by x+6. Determine what value C must equal if x+6 is a factor of p(x).

a)

C would need to be equal to 0.

b)

C would need to be equal to 12.

c)

C would need to be equal to -12.

d)

C would need to be equal to 300.

Answer Key

Remainder Theorem and Factor Theorem

Total questions: 20

1.
a)

Yes, x-3 is a factor of p(x) since the remainder is 0.

2.
a)

x-2 is not a factor of x2−3x+5x^2-3x+5   since the remainder is 3.

3.
b)

2 is not a solution of x2−3x+5x^2-3x+5   since the remainder is 3.

4.

3/2 is a solution of p(x).

 - 

2x - 3 is a factor of p(x).

, 

1 is a solution of p(x).

 - 

x-1 is a factor of p(x).

, 

3 is a solution of p(x).

 - 

x - 3 is a factor of p(x).

, 

0 is a solution of p(x).

 - 

x is a factor of p(x).

5.

x + 3 is a factor of p(x).

 - 

-3 is a solution of p(x).

, 

x - 5 is a factor of p(x).

 - 

5 is a solution of p(x).

, 

3x - 5 is a factor of p(x).

 - 

5/3 is a solution of p(x).

, 

x + 5 is a factor of p(x).

 - 

-5 is a solution of p(x).

6.
b)

1 is not a solution of p(x)

7.

is a factor of 5x2−10x−405x^2-10x-40  

 - 

x + 2...

, 

is a solution of 5x2−10x−405x^2-10x-40  

 - 

-2...

, 

0

 - 

The remainder is...

, 

x+2

 - 

The work provided shows the polynomial 5x2 - 10x - 40 being divided by...

8.
a)

x-7 is a factor of p(x).

, d)

7 is a solution of p(x).

9.
b)

x+6 is not a factor of x3−x2−40x +13x^3-x^2-40x\ +13  .

10.
a)

5 is a solution of p(x) since the remainder is 0.

11.
a)

x-5 is a factor of p(x) since 5 was a solution.

12.
c)

x+4 is not a factor of x2+16x^2+16  

13.
a)

True

14.
c)

-2 is not a solution of p(x) since the remainder is equal to 33.

15.
a)

The remainder is 18.

16.
b)

3 is not a solution of 2x3−15x+92x^3-15x+9  .

17.
a)

x-3 is not a factor of 2x3−15x+92x^3-15x+9  .

18.
b)

x-4

19.
b)

The x-value 2

20.
b)

C would need to be equal to 12.

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FAQs

What does the Remainder Theorem and Factor Theorem worksheet cover?

This worksheet helps Grade 10–11 students use polynomial and synthetic division remainders to determine whether a number is a solution and whether a corresponding linear expression is a factor. Its 20 items combine multiple choice, matching, and multiple select, requiring students to connect a zero remainder with statements such as “5 is a solution” and “x - 5 is a factor,” while interpreting nonzero remainders as evidence that the factor relationship does not hold.

How can I use this worksheet to teach the connection between remainders, factors, and solutions?

Begin by modeling one synthetic-division example and have students state what the remainder means before showing the answer choices. Then use the matching items to connect a solution value, such as 3 or 3/2, with its related factor, and finish with the multiple-select item to discuss why both a factor statement and a solution statement can be correct. Ask students to explain how the sign in x - a corresponds to the value a.

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

Watch for students treating any remainder as proof of a factor or solution instead of checking whether the remainder is zero. Students may also reverse the sign relationship, matching x + 7 with 7 instead of -7, or confuse a solution value with the factor expression itself. In the multiple-select item, check that students select both valid consequences rather than stopping after identifying only the factor or only the solution.

How do I assign and grade this worksheet?

This worksheet is available as a printable PDF and as a digital quiz on Wayground, with a complete answer key for checking all 20 items. Printable submissions can be graded by scanning student work with the Wayground for Teachers app. The printable option can also support schools that are reducing screen time.

How can I differentiate this polynomial remainders worksheet for my students?

Use the worksheet’s Advanced Settings to adjust font spacing or increase the font size for the synthetic-division and matching items. You can also enable a dyslexia-friendly font or translate the worksheet into another language, depending on student needs. These changes preserve the worksheet’s multiple-choice, matching, and multiple-select format.

Where can I find more worksheets like this on Wayground?

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

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