Category Archives: Algebra

For number pairs $(r_i, c_i)$, $i=1,2,…,n$, where $n\ge 1$, $r_i\ge 0$ and $c_i\ge 0$, they have the following property:$$\sum_{i=1}^{n}r_i=\sum_{i=1}^{n}c_i=n^2$$ Additionally, there exists a positive value of $k$ so that for every $i$ value,$i=1,2,…,n$, the following three inequalities hold:$$r_i\le \dfrac{n^3}{k}\ \ … Continue reading →

Prove that there is no polynomial $P(x)$ with integer coefficients such that $$P(\sqrt[3]{5}+\sqrt[3]{25})=2\sqrt[3]{5}+3\sqrt[3]{25}$$🔑 Proof: Assume that there is a polynomial $P(x)$ of degree $n$, with integer coefficients, and without trivial root of $0$, such that$$P(\sqrt[3]{5}+\sqrt[3]{25})=2\sqrt[3]{5}+3\sqrt[3]{25}$$Therefore, $P(x)$ can be expressed as … Continue reading →

Problem1: Let $x_{1}$ and $x_{2}$ be the root of $x^2-7x-9=0$. Find the value of $|x_{1}-x_{2}|$. Solution $$|x_{1}-x_{2}|=\dfrac{\sqrt{b^2-4ac}}{|a|}=\dfrac{\sqrt{(-7)^2-4\cdot(-9)}}{|1|}=\sqrt{85}$$ Problem 2: Find all solutions of $\sqrt{x+10}-\dfrac{6}{\sqrt{x+10}}=5$. Solution Let $y=\sqrt{x+10}$, we have $$y-\dfrac{6}{y}=5$$ Multiplying $y$ on both side of the equation, we have … Continue reading →