# Interesting problems concerning factorization and simplifictions of expressions

I feel that deep within the knowledge of many users here dwell great problems regarding clever tricks for simplifying problems, and factoring algebraic expressions. I would be very much interested in seeing a large list of problems, regarding such problems.

Any problem that is clever, has something to do with factorization or simplification should be placed here. Now, I would hope that only the crème de la crème. The very best of such problems. Not just a mass of standard ones.

Any problem that does not require advanced math such as, topology, linear algebra, differential equations are allowed or imaginary numbers. The problem may indeed be very difficult, but it should not consist of topics beyond the first year of college.

Does anyone have any similar problems, or expressions like the small list I compiled below? (My favourite problem of the answers, will get the reward.)

1. If possible simplify these expressions into simpler terms
2. If further simplification is not possible, factor the expressions as much as possible in the reals.

• $$\qquad \frac{7}{\sqrt{7}}$$
• $$\qquad 2x^2 -1 + 2x^2$$
• $$\qquad x(x-1)-2(x-1)$$
• $$\qquad x^2+3x-2x-6$$
• $$\qquad \frac{t^2-6t+9}{t^2-8t+15}$$
• $$\qquad \frac{t^2 - 2t - 4}{2t + \sqrt{2}}$$
• $$\qquad \frac{1}{2} \ln \left( \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\right)$$
• $$\qquad \big(\cos(x) + \sin(x)\big)^2+\big( \cos(x) - \sin(x) \big)^2$$
• $$\qquad \frac{3 - 4\sqrt{3}x}{\sqrt{3}}$$
• $$\qquad \frac{1-4y^2}{6y-3}$$
• $$\qquad \frac{\cos x \cdot (\sin x+1)^2 - \cos \cdot (\sin x-1)^2}{(\sin x)^4 - (\cos x)^4}$$
• $$\qquad \sqrt{4 - 2\sqrt{3}}$$
• $$\qquad \frac{2t^2-1}{2t + \sqrt{2}}$$
• $$\frac{2^{2x-1} - 2^{x-1}}{2^{2x-1}}$$
• $$\frac{\qquad\dfrac{5p+10}{p^2-4}\qquad}{\dfrac{3p-6}{(p-2)^2}}$$
• $$\sqrt[3]{\frac{x^3-6x^2+12x-8}{x^3+3x^2+3x+1}}$$
• $$y^2 - 4 - x^2 + 4x$$
• $$\sqrt{9x^2-6x+1}$$
• $$\ln \left( \sqrt{x-1} \right) \exp \left( \ln(4) + \ln \left( \frac{1}{2} \right)\right)$$
• $$\sqrt[\Large 4]{\dfrac{6 - 2\sqrt{5}}{6 + 2\sqrt{5}}}$$
• $$\dfrac{\left( 1 + \dfrac{1}{\sqrt[4]{x}}\right) (x-1)}{\left( 1 + \dfrac{1}{\sqrt{x}}\right)}$$
• $$\qquad \dfrac{x^{\frac{3}{2}} \cdot \sqrt[2]{\frac{y}{x} \cdot }\left( x^2 - 2x y^3 + y^6 \right) }{\left( \sqrt{x} - \sqrt{y^3} \right) x \left( \sqrt{x} + \sqrt{y^3} \right) }$$
• $$\qquad \Large 2^{\frac{\log\left( \frac{100}{x}\right) - 1 }{- \log(2)}}$$
• $$x^6-2x^3+1$$
• $$x^6+3x^4+3x^2+1$$
• $$x^3+x^2-x-1$$
• $$(2k+1)^8-1$$
• $$x^3 + 1$$
• $$x^4 + 1$$
• $$x^4 + x^2 + 1$$
• $$\sqrt{18(\sqrt[3]10 - 2)}$$
• $$\frac{n! + (n-1)n!}{(n-2)!}$$
• $$\qquad \sqrt{12 + 5 \sqrt 6}$$
• $$\sqrt{\frac{1}3 \sqrt{6} (12 + 5\ \sqrt 6)}$$
• $$x^4-6x^3+11x^2-6x+1$$
• $$x^4+6x^3-5x^2-10x-3$$
• $$\sqrt{1 + \sin 2x}$$
• $$x(x+1)(x+2)(x+3) - 120$$
• $$\sqrt{ \frac{1}{1 + \sin x } }$$
• $$\qquad \frac{\sin t + \sin 3t}{\cos 3t + \cos t}$$
• $$\sqrt[3]{26+15\sqrt{3}}$$
• $$x^5+x+1$$
• $$\sqrt{2^{6/7}}$$

Regards werner

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Why don't you just post your own simplifications as an answer? – Peter Phipps Dec 11 '11 at 15:47
Are you saying to post the answer to these problem as an answer, or the problems themselves as an anwer? I merely want more answers like these – N3buchadnezzar Dec 11 '11 at 16:23
I don't understand why you are asking question 1 when it seems that you know what the "answers" are before hand. – Peter Phipps Dec 11 '11 at 18:15
The only question i proposed was if people would post similar problems to the ones I posted. When one posts equations one must adress, what is supposed to be done with these. Whichis what I wrote in 1 and 2. So the answers I am looking for, are answers that can be solved following 1 and 2. If you know a cleared way to state this, feel free to edit my post =) – N3buchadnezzar Dec 11 '11 at 18:27
This looks like a good candidate for a [big-list] CW sort of question... – Asaf Karagila Dec 12 '11 at 7:02

$$2x^2 y^2 + 2x^2 z^2 + 2y^2 z^2 - x^4 - y^4 - z^4 = (x+y+z)(x+y-z)(x-y+z)(-z+y+z)$$
Your expression involving a quotient of sines and cosines is an instance of the tangent-half-angle formula in the form $$\frac{\sin\alpha + \sin\beta}{\cos\alpha+\cos\beta} = \tan\frac{\alpha+\beta}{2}.$$
If you want to include trigonometric identities in your list, look at Wikipedia's list of trigonometric identities. Some of the items there are routine things; others are exotic. As an example that's only a little bit exotic, everybody raise your hands if you knew that $$\sec(\theta_1 + \cdots + \theta_n) = \frac{\sec\theta_1 \cdots \sec\theta_n}{e_0 - e_2 + e_4 - \cdots}$$ where $e_n$ is the $n$th-degree elementary symmetric polynomial in the variables $\tan\theta_1,\ldots,\tan\theta_n$. Or the fact that if $x+y+z=\pi$ then $\sin(2x)+\sin(2y)+\sin(2z)= 4\sin x\sin y\sin z$, which I think is not well known in some parts of the world.