# Does anyone know of any good problems on factorization?

1. Dec 11, 2011

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

--------------------------------------------------------

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

--------------------------------------------------------

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
. =)