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5B Long Division and Synthetic Division - Printable Mathematics Worksheets - Wayground

5B Long Division and Synthetic Division

14 questions

10th - 10th Grade

Mathematics

This Grade 10 mathematics worksheet, 5B Long Division and Synthetic Division, gives students 14 multiple-choice questions on dividing polynomials and interpreting the results. Students identify quotients and remainders, use synthetic division to divide polynomials, determine whether a binomial is a factor, apply the Remainder Theorem by evaluating p(a), and arrange coefficients in descending order with zero placeholders for missing terms. The questions also check essential division vocabulary and procedural accuracy, including recognizing an incorrect worked example. This free printable worksheet is designed for algebra practice, review, or an assessment of students’ readiness to use synthetic division with polynomials such as x³, x⁴, and x⁶. A complete answer key is included so teachers can quickly review students’ choices and address errors in setup, coefficient order, or remainder calculations.

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Worksheets

5B Long Division and Synthetic Division

Total questions: 14

Name
Class
Date
1.
a)
x - 7
b)
x2 - 7
c)
x + 7
d)
x2 + 3x - 54
2.
a)
2x3 + 3x + (7/x-4)
b)
2x2 + 3x + 8 + (7/x-4)
c)
2x2 + 3x + 8
d)
2x2 - x + 10 + (6/x-4)
3.
            Is (x-2) a factor of             f(x)= x3-8x2+14x-4?
a)
Yes, (x-2) is a factor. There is a remainder.
b)
No, (x-2) is  not a factor. The remainder is zero.
c)
Yes, (x-2) is a factor. The remainder is zero.
d)
No, (x-2) is  not a factor. There is a remainder. 
4.
Divide using synthetic division 
(2x³ + 4x² - 5) by (x + 3).
a)
2x² + 2x - 4  R(-12)
b)
2x² - 2x + 6  R(-23)
c)
2x² - 2x + 8  R(-20)
d)
2x² - 4x + 1  R0
5.
a)
x + 5 + (2/2x - 4)
b)
x - 8
c)
x - 5 + (3/2x - 4)
d)
x + 5
6.
The answer to a division problem is known as the...?
a)
product
b)
sum
c)
difference
d)
quotient
7.
What does it mean to be a factor of a number?
a)
Divides evenly with a remainder of zero.
b)
Divides unevenly with a remainder of 1.
8.

If you are missing a term in your polynomials (both dividend and divisor), you must put a zero as a coefficient place holder for that term.

a)

True

b)

False

c)

Only for the dividend

d)

Only for the divisor

9.
If 𝒙2+𝟑𝒙+𝟕 is divided by  𝒙+𝟐, then what is the remainder?
a)
0
b)
-3
c)
9
d)
5
10.
When p(x) = 9x4- 45x3 + 37x2 + x +2 is divided by x - 2, a student can determine the remainder by evaluating p(2). What is the value of p(2)?
a)
p(2) = 656
b)
p(2) = 652
c)
p(2) = -64
d)
p(2) = -368
11.
What should be the order of the polynomial coefficients if
(3x - 4x3 + 6x4 + 1) / (x + 3)
a)
3   0   -4   6   1
b)
6   -4   3   1   0
c)
6   -4   0   3   1
d)
-6   4   0   -3   -1
12.
If you were dividing x6 + 4x3 + 2, how many 0's would you need when setting up the top row of your synthetic division?
a)
0
b)
2
c)
4
d)
6
13.
What is the quotient when (10n2 - 37n - 12) is divided by (n - 4)?
a)
(10n - 3)
b)
(3n - 10)
c)
(10n + 3)
d)
(3n + 10)
14.
Is this division problem worked correctly?
a)
This is correct!
b)
This is incorrect!
Answer Key

5B Long Division and Synthetic Division

Total questions: 14

1.
a) x - 7
2.
b) 2x2 + 3x + 8 + (7/x-4)
3.
c) Yes, (x-2) is a factor. The remainder is zero.
4.
b) 2x² - 2x + 6  R(-23)
5.
d) x + 5
6.
d) quotient
7.
a) Divides evenly with a remainder of zero.
8.
a)

True

9.
d) 5
10.
c) p(2) = -64
11.
c) 6   -4   0   3   1
12.
c) 4
13.
c) (10n + 3)
14.
b) This is incorrect!

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FAQs

What does the 5B Long Division and Synthetic Division worksheet cover?

This Grade 10 mathematics worksheet uses 14 multiple-choice questions to assess polynomial division with synthetic division. Students find quotients and remainders, test whether a binomial is a factor, evaluate p(a) to find a remainder, order coefficients correctly, and insert zero placeholders for missing polynomial terms.

How can I use this worksheet to teach synthetic division and the Remainder Theorem?

Introduce the coefficient-row setup first, emphasizing descending powers and zero placeholders for missing terms. Then model one synthetic division problem and connect its final value to the Remainder Theorem, using the worksheet’s factor and p(2) questions for guided discussion before students complete the remaining multiple-choice items independently.

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

Watch for students omitting zero coefficients in polynomials such as x⁶ + 4x³ + 2 or placing coefficients in the wrong order before beginning synthetic division. Students may also confuse the quotient with the remainder, use the wrong synthetic value for a divisor such as x + 3, or conclude that a binomial is a factor when the remainder is not zero.

How do I assign and grade this long division and synthetic division worksheet?

The worksheet is available as a printable PDF and as a digital quiz on Wayground, and it includes a complete answer key for the 14 multiple-choice questions. Printable submissions can be graded by scanning student work with the Wayground for Teachers app, while the digital version can be assigned through the platform for online completion.

How can I differentiate this synthetic division worksheet for my students?

Under Advanced Settings, you can create an alternate version of this worksheet by changing font spacing or font size to make the polynomial expressions and answer choices easier to read. You can also apply a dyslexia-friendly font or translate the worksheet into another language while keeping its synthetic division, remainder, and coefficient-order questions.

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.

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