Printable pre-algebra, algebra and geometry worksheets

Rational Expression Worksheets Generator

Make printable practice on rational expressions – simplify, multiply, divide, add and subtract, then solve rational equations – with answers in factored form.

Open on EasyMathPrint

Embed

Embed this tool

Copy this code into your website, blog or class page. The tool runs inside your page.

Please keep the credit link below the tool.

This free rational expressions worksheet generator makes printable practice sheets on simplifying, multiplying, dividing, adding and subtracting rational expressions, and solving rational equations. It runs in your web browser with no sign-up, and the problems are made on your device, so nothing is uploaded. Use it to build simplifying rational expressions worksheets for class, homework, or your own practice. Each sheet can include an answer key with every answer in simplest factored form, and you can print it or save it as a PDF. Easy problems use single terms such as 15x²/(12x); Hard problems use trinomials and opposite factors such as (3 − x). Everything you make here is free to use in class or in products you sell, such as Teachers Pay Teachers resources, Etsy printables or KDP workbooks.
Rational expression worksheets example

How to use the Rational Expressions Worksheet Generator

  1. In the Problem types list on the left, tick one or more types. Simplify, Multiply and Divide are ticked when the page opens.
  2. Under Difficulty, choose Easy, Medium or Hard.
  3. Under Number of problems, choose 6, 8, 10, 12, 15, 20, 24 or 30. If you ticked several types, the problems are shared evenly between them.
  4. Open More options to set the Columns (Automatic, 1, 2 or 3), the Paper size (Letter or A4), a Worksheet title, and the Name and date line checkbox.
  5. Keep Answer key on a separate page ticked if you want a key. It is on by default.
  6. Check the preview on the right, which shows the sheet as it will print. Press Show answers to see each answer under its problem, and press it again to hide them.
  7. Press New worksheet for a fresh set of problems with the same settings.
  8. Press Print / Save as PDF. In the print dialog, choose “Save as PDF” if you want a file.

Press Copy link to copy a link that reopens exactly the same worksheet. The settings are remembered in your browser for your next visit. The tool also works on phones and tablets, but printing needs a printer or the “Save as PDF” option.

How do you simplify rational expressions

A rational expression is a fraction whose numerator and denominator are polynomials. To simplify one, factor the numerator and the denominator completely, then cancel the factors they share. Only factors can be cancelled, never terms, so you cannot cancel the x in (x + 3)/x.

For 5x³/(20x²), factor both parts: 5·x·x·x over 5·4·x·x. Cancel 5 and x², leaving x/4. Watch for sign changes. A factor such as (3 − x) equals −(x − 3), so (2x² + 7x − 15)/(2(3 − 2x)) factors to (2x − 3)(x + 5) over −2(2x − 3), which simplifies to −(x + 5)/2.

How do you multiply and divide rational expressions

To multiply, factor every numerator and denominator, cancel any factor that appears on both a top and a bottom, then multiply the remaining tops and bottoms. For (9)/(6x) · (4x)/(6x), the product is 36x/(36x²), which simplifies to 1/x.

To divide, multiply by the reciprocal of the second expression, then follow the multiplication steps. For 10/(8x²) ÷ x²/12, rewrite it as 10/(8x²) · 12/x². That gives 120/(8x⁴), which simplifies to 15/x⁴.

Simplifying rational expressions in this way is also called reducing rational expressions. Your final answer should have no common factors left.

How do you solve a rational equation?

Multiply both sides by the least common denominator to clear the fractions, solve the equation that is left, then check every solution against the original denominators. A value that makes a denominator zero is an extraneous solution and must be rejected.

For (x² − 6x + 16)/(x − 4) = 8/(x − 4), multiplying by x − 4 gives x² − 6x + 16 = 8, so x² − 6x + 8 = 0 and (x − 2)(x − 4) = 0. The value x = 4 makes the denominator zero, so the only solution is x = 2. The answer key on Hard sheets writes this as “x = 2 (x = 4 is extraneous)”, and when every candidate is rejected it says there is no solution.

Sample problems from this generator

  1. 3(x − 1)(x + 1)2x − 2 Answer: 3(x + 1)2
  2. x − 63x(x − 6) Answer: 13x
  3. 4(x − 2)6(x − 2) · 6x(x − 3)x(x − 2) Answer: 4(x − 3)x − 2
  4. 3x(x − 2)4(x + 7) ÷ x − 24(x − 4) Answer: 3x(x − 4)x + 7

Frequently asked questions

What are excluded values?

Excluded values are the x-values that make the original denominator zero, because division by zero is undefined. For 1/(2x + 6), set 2x + 6 = 0 to get x = −3, so the answer is x ≠ −3. Find them from the original denominator, before you cancel anything.

How do you add rational expressions with different denominators?

Rewrite each expression over the least common denominator (LCD), combine the numerators, then simplify. For 1/x − 2/x², the LCD is x². That gives x/x² − 2/x², which equals (x − 2)/x². If the denominators already match, you only combine the numerators.

How do you solve rational equations?

Multiply both sides by the LCD, solve the resulting equation, and check each answer. Reject any solution that makes a denominator zero, because it is an extraneous solution. Always test your answers in the original equation.

Is there an answer key?

Yes. The Answer key on a separate page option is on by default. When you print, the key starts on a new page after the worksheet and lists every problem number with its answer. To check answers on screen first, press Show answers.

Can I make several versions of the same worksheet?

Yes. Each click on New worksheet gives different problems with the same settings. No problem repeats on one sheet, so you can print several versions, for example one for each row of desks.

Is the generator free to use?

Yes. It is free, requires no sign-up, and runs in your browser. Nothing you create is uploaded.

Can I sell the rational expressions worksheets I make here on Teachers Pay Teachers?

Yes. Everything you make here is royalty-free for personal, classroom and commercial use, including Teachers Pay Teachers, Etsy printables, KDP workbooks and print-on-demand products. No credit is required. The only thing you may not do is copy the generator or the site itself to run a competing service. See the terms.