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I'm searching for exercises for practising operations with terms. They should involve

  • working with decimal numbers and fractions (ideally one should convert decimal numbers to simple fractions like halves, fourths or fifths to simplify calculations and be able to cancel out factors)
  • expanding products and getting rid of parenthesis, in particular deal with plus and minus signs and possibly use binomial formulas
  • using power laws
  • combining terms

and ideally give a possibly simple result. An example of what I have in mind would be something along the lines of

$$1.25(6a-4b)^2(0.4a^2b)^2 - 2.2(5a-4b)(4a-5b)(0.5ab^2)(2.5a^3b)$$

(however, this example doesn't simplify well at all and gives many terms with ugly coefficients).

Hints on how to design such problems myself or even automatize the generation of such problems with a little program are also welcome!

Thanks for any input!

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    $\begingroup$ If you haven't seen Kuta software, it can generate similar problems to these. However, I'm not sure if they can include all of the properties in one problem. Try Kuta's "Infinite Algebra 2" kutasoftware.com/freeia2.html or "Infinite Precalculus" (accessible from same link). $\endgroup$
    – Opal E
    Commented Mar 28, 2019 at 18:20
  • $\begingroup$ Thanks for the hint, but as you said, they don't include all the nice properties, I fear, the problems are way too simple (I tried "Simplifying algebraic expressions" and "Multi-step equations")... $\endgroup$
    – Photon
    Commented Mar 28, 2019 at 18:25
  • $\begingroup$ Generating "complex" problems like this is not trivial, especially since the programer would need to offer up a ton of topics (like the Kuta Software people above). They'd also need to make sure that no other site could sneakily reuse the code to produce worksheets on a competing site, using CAPTCHAs and accounts and verification and so on. Here, polynomials, up to degree x, with y terms, using z% fractions, w% decimals, u% of terms using the distributive property over expressions of degree v, with solutions that are s% integer coefficients, t% fractional coefficients, with...gets complex! $\endgroup$
    – Sciolism Apparently
    Commented Mar 31, 2019 at 19:37
  • $\begingroup$ If you have a "simple expression" of a polynomial $P(a,b)$ and you want to make it "more complicated", you can find an invertible function $f$ such that $(a,b)=f(c,d)$ and $(c,d)=f^{-1}(a,b)$. Let $Q=P\circ f$ then the expression you want is $\left(Q\circ f^{-1}\right)(a,b)$. $\endgroup$
    – user5402
    Commented Apr 1, 2019 at 17:33
  • $\begingroup$ @BPP: I think, other than in abstract maths, finding a particular example is more difficult than proving existence here. ;) $\endgroup$
    – Photon
    Commented Apr 1, 2019 at 20:50

2 Answers 2

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It costs $5/month (for educators) to use Wolfram Alpha in its practice worksheets model. It will generate a lot of problems for you, but I'm not 100% sure it gives you the granularity you want. I really like it.

https://www.wolframalpha.com/pro-for-educators/

Also, Math.com has a worksheet generator, which allows some specification of fraction use and difficulty of problems, etc.

http://www.math.com/students/worksheet/algebra_sp.htm

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Something like this?

$$a. \left({a+b \over a-b} + {a-b \over a+b}\right) \div \left({a^2 \over a^2-b^2} + {1 \over {a^2 \over b^2}-1}\right)$$

$$b. \left({x^2y - xy^2 \over x-y} + xy\right) \times \left({y \over x} + {x \over y}\right)$$

$$c. \left({n \over m-n} + {m \over {m + n}} \right) \times \left({m^2 \over n^2} + {n^2 \over m^2} - 2\right)$$

$$d. \left({x + 1 \over x - 1} - {x - 1 \over {x + 1}} + 4x \right) \times \left(x - {1 \over x} \right)$$

$$e. \left(1 + {a \over x} + {a^2 \over {x^2}} \right) \left(1 - {a \over x} \right) \times {x^3 \over {a^3 - x^3}} $$

$$f. {4x - 3 \over 3 - 2x} - {4 + 5x \over {3 + 2x}} - {3 + x - 10x^2 \over {4x^2 - 9}} $$

There should be plenty of exercises like these in any decent textbook, I pulled these from a 7th grade Russian algebra textbook. Start from Chapter 4, page 59 for monomials, and scroll forward for polynomials, factoring, completing the square, etc. Some exercises have answers in the back of the book.

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