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Polynomial Division

Practise dividing one algebraic expression by another in this set of exercises.

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This is level 5: mixed exercise from old textbook. This long, difficult exercise is only included here as a challenge for the very adventurous. The real challenge is being able to type in the answers so that they exactly match the answers given at the back of the ancient textbook this exercise was taken from (A First Book in Algebra, by Wallace C. Boyden written in 1895)



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$$\frac{x^6 - 5x^3 + 3 + 5x^4 - 10x - x^5 + 10x^2}{x^2 + 3 -x}$$

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$$\frac{x^6 - 2x^3 - 2 + x - 3x^5 + 2x^4 - 5x^2}{x^3 + 2 + x}$$

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$$\frac{a^5 - a - 2a^2 - a^3}{a + a^3 + a^2}$$

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$$\frac{x^6 - 2x^3 - x^2 - x^4}{1 + x^2 + x}$$

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$$\frac{a^{11} - a^2}{a^3 - 1}$$

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$$\frac{a^{12} - a^4}{a^2 + 1}$$

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$$\frac{x^4 - x^2y^2 - 2xy^3 + 2y^4}{x^2 - 2xy + y^2}$$

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$$\frac{4a^4 + 81b^4}{2a^2 + 6ab + 9b^2}$$

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$$\frac{\frac{1}{2}x^4 + \frac{3}{4}x^3y - \frac{1}{3}x^2y^2 + \frac{7}{6}xy^3 - \frac{1}{3}y^4}{x^2 - \frac{1}{2}xy + y^2}$$

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$$\frac{\frac{2}{9}y^3 - \frac{5}{36}x^2y + \frac{1}{6}xy^2 + \frac{1}{6}x^3}{ \frac{1}{2}x + \frac{1}{3}y}$$

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$$\frac{\frac{1}{4}(x - y)^5 - (x - y)^3 - \frac{1}{2}(x - y)^2 -\frac{1}{16}(x - y)}{\frac{1}{2}(x - y)^2 + (x - y) +\frac{1}{4}}$$

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$$\frac{16x^8 - 81y^4}{27y^3 + 18x^2y^2 + 8x^6 + 12x^4y}$$

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$$\frac{4x^4 - 10x^2 + 6}{x + 1}$$

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$$\frac{4a^4 - 5a^2b^2 + b^4}{2a - b}$$

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$$\frac{a^6 - b^6 + a^4 + b^4 + a^2b^2}{a^2 - b^2 + 1}$$

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$$\frac{x^6 - y^6 - x^4 - y^4 - x^2y^2}{x^2 - y^2 - 1}$$

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$$\frac{81a^{12} - 16b^8}{12a^3b^4 - 8b^6 - 18a^6b^2 + 27a^9}$$

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Did you know that there is a method for finding the remainder without having to do long division? It is called:

The Remainder Theorem

If a polynomial \(f(x)\) is divided by \((x-a)\) then the remainder is \(f(a)\)

Do some independent research to find out more about this theorem and how it can be used to complete this exercise really quickly.

You may be interested to know that students were answering these very same questions over one hundred years ago. This exercise comes from a textbook written in the 1890s.

This is Polynomial Division level 5. You can also try:
Level 1 Level 2 Level 3 Level 4


Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

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Description of Levels



Algebraic Fractions - For a simpler introduction have a go at Algebraic Fractions level 1.

Level 1 - Divide a polynomial by a single term

Level 2 - Divide a polynomial by a linear expression

Level 3 - Find the remainder

Level 4 - Divide a polynomial by a quadratic or cubic expression

Level 5 - Mixed exercise from old textbook

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Algebra including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Divident ÷ Divisor = Quotient

To type indices or exponents use the up arrow key ^ then type the number followed by the right arrow. The terms in your answer should be in the same order as the terms in the question.

Long Division Example

Algebraic long division example

Beth's Method (see video above)

Algebraic long division example

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

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