Does anyone know of any good problems on factorization?

[Nebuchadnezza]

Now, for my students and fellow teachers. I am looking to collect a great amount of problems involving factorization, and simplifications of problems.

Below is a smal portion of the type of problems I am looking for.

"Rules")

1) Simplify a problem, until it can not be simplified anymore.
2) If a problem is not possible to simplify anymore, try to factor it.

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$$1) \qquad \frac{7}{\sqrt{7}}$$
$$2) \qquad 2x^2 -1 + 2x^2$$
$$3) \qquad x(x-1)-2(x-1)$$
$$4) \qquad x^2+3x-2x-6$$
$$4.1) \qquad \frac{t^2-6t+9}{t^2-8t+15}$$
$$4.2) \qquad \frac{t^2 - 2t - 4}{2t + \sqrt{2}}$$
$$5) \qquad \frac{1}{2} \ln \left( \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\right)$$
$$6) \qquad \big(\cos(x) + \sin(x)\big)^2+\big( \cos(x) - \sin(x) \big)^2$$
$$7) \qquad \frac{3 - 4\sqrt{3}x}{\sqrt{3}}$$
$$8) \qquad \frac{1-4y^2}{6y-3}$$
$$9) \qquad \frac{\cos x \cdot (\sin x+1)^2 - \cos \cdot (\sin x-1)^2}{(\sin x)^4 - (\cos x)^4}$$
$$10) \qquad \sqrt{4 - 2\sqrt{3}}$$
$$10.1) \qquad \frac{2t^2-1}{2t + \sqrt{2}}$$
$$11) \qquad \frac{2^{2x-1} - 2^{x-1}}{2^{2x-1}}$$
$$12) \qquad \frac{\qquad\dfrac{5p+10}{p^2-4}\qquad}{\dfrac{3p-6}{(p-2)^2}}$$
$$13) \qquad \sqrt[3]{\frac{x^3-6x^2+12x-8}{x^3+3x^2+3x+1}}$$
$$14) \qquad y^2 - 4 - x^2 + 4x$$
$$15) \qquad \sqrt{9x^2-6x+1}$$
$$16) \qquad \ln \left( \sqrt{x-1} \right) \exp \left( \ln(4) + \ln \left( \frac{1}{2} \right)\right)$$
$$17) \qquad \sqrt[\Large 4]{\dfrac{6 - 2\sqrt{5}}{6 + 2\sqrt{5}}}$$
$$18) \qquad \dfrac{\left( 1 + \dfrac{1}{\sqrt[4]{x}}\right) (x-1)}{\left( 1 + \dfrac{1}{\sqrt{x}}\right)}$$
$$19) \qquad \dfrac{x^{\frac{3}{2}} \cdot \sqrt[2]{\frac{y}{x} \cdot }\left( x^2 - 2x y^3 + y^6 \right) }{\left( \sqrt{x} - \sqrt{y^3} \right) x \left( \sqrt{x} + \sqrt{y^3} \right) }$$
$$20) \qquad \Large 2^{\frac{\log\left( \frac{100}{x}\right) - 1 }{- \log(2)}}$$
$$21) \qquad x^6-2x^3+1$$
$$22) \qquad x^6+3x^4+3x^2+1$$
$$23) \qquad x^3+x^2-x-1$$
$$24) \qquad (2k+1)^8-1$$
$$25) \qquad x^3 + 1$$
$$26) \qquad x^4 + 1$$
$$27) \qquad x^4 + x^2 + 1$$
$$28) \qquad \sqrt{18(\sqrt[3]10 - 2)}$$
$$29) \qquad \frac{n! + (n-1)n!}{(n-2)!}$$
$$30) \qquad \sqrt{12 + 5 \sqrt 6}$$
$$31) \qquad \sqrt{\frac{1}3 \sqrt{6} (12 + 5\ \sqrt 6)}$$
$$32) \qquad x^4-6x^3+11x^2-6x+1$$
$$33) \qquad x^4+6x^3-5x^2-10x-3$$
$$34) \qquad \sqrt{1 + \sin 2x}$$
$$35) \qquad x(x+1)(x+2)(x+3) - 120$$
$$37) \qquad \sqrt{ \frac{1}{1 + \sin x } }$$
$$37.1) \qquad \frac{\sin t + \sin 3t}{\cos 3t + \cos t}$$
$$38) \qquad \sqrt[3]{26+15\sqrt{3}}$$
$$39) \qquad x^5+x+1$$
$$40) \qquad \sqrt{2^{6/7}}$$

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I would also want some opinions on these. Of course I am not asking anyone to solve any of these
, just feedback if they are good or bad problems.

Do anyone here know any similar problems regarding fun problems to simplify or factor?
It can be anything from elementary algebra, logarithms, powers, factorials, etc. =)

 

[jvicens]

Hello! As a fellow scientist and teacher, I believe that these are great problems for students to practice their skills in simplification and factorization. They cover a wide range of mathematical concepts and can challenge students at different levels. I particularly like the problems involving logarithms and powers, as those can be tricky for students to understand at first.

In addition to these problems, I would suggest incorporating real-life scenarios or applications into some of the problems. This can make the problems more interesting and relevant for students. For example, you can ask students to simplify or factor an equation that represents the growth of a population over time, or the amount of interest earned on a loan. This can help students see the practical applications of these mathematical skills.

Overall, I think these are great problems and I'm sure your students will benefit from practicing them. Keep up the good work!