Algebra

Polynomial Arithmetic

Polynomial Arithmetic: Level 3 Challenges

         

640003+64000+3(1640)+13\sqrt{\sqrt{\sqrt[3]{\color{#3D99F6}{64000}} + {\sqrt[3]{\color{#3D99F6}{64000} + \color{teal}{3(1640) + 1}}}}}

With the assistance of the algebraic identity (a+b)3=a3+b3+3ab(a+b) (a+b)^3 = a^3 + b^3 + 3ab(a+b) , evaluate the above expression.

If aa and bb are non-zero real numbers, simplify: (a+1a)2+(b+1b)2+(ab+1ab)2(a+1a)(b+1b)(ab+1ab)\left(a + \frac1a\right)^2+\left(b + \frac1b\right)^2 + \left(ab + \frac1{ab}\right)^2 \\ - \left(a + \frac1a\right)\left(b + \frac1b\right)\left(ab + \frac1{ab}\right)

A lazy mathematician believes that

a2+b2=(a+b)2.a^2 + b^2 = (a + b)^2.

If aa and bb are both integers chosen from the interval [100,100]\left[-100, 100\right], then find the number of ordered pairs (a,b)(a, b) that satisfy the equation above.

1+x3+1x3=53\sqrt[3]{ 1+ \sqrt{x}} + \sqrt[3]{1 - \sqrt{x}} = \sqrt[3]{5}

One solution of the above equation is of the form ab,\frac{a}{b}, where aa and bb are coprime positive integers.

Find a+ba+b.

2x2y=14x4y=53xy=? \begin{aligned} \large \color{#3D99F6}2^{\color{#624F41}x} - \color{#3D99F6}2^{\color{#624F41}y} & = & \large 1 \\ \\ \large \color{#20A900}4^{\color{#624F41}x}- \color{#20A900}4^{\color{#624F41}y} & = & \large { \frac 5 3} \\ \\ \\ \large {\color{#624F41}x} - {\color{#624F41}y} & = & \large \, ? \end{aligned}

Details and Assumptions:

  • xx and yy are real numbers.
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