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如果你也在线性代数linearalgebra这个学科遇到相关的难题,请随时添加vx号联系我们的代写客服。我们会为你提供专业的服务。 linearalgebra™长期致力于留学生网课服务,涵盖各个网络学科课程:金融学Finance,经济学Economics,数学Mathematics,会计Accounting,文学Literature,艺术Arts等等。除了网课全程托管外,linearalgebra™也可接受单独网课任务。无论遇到了什么网课困难,都能帮你完美解决! 1 Algebra 1.5 Products, Factorizations and Telescoping Sums**Basic Formulae**1. Expand \((a+b)^{2}\). 2. Expand \((a-b)^{2}\). 3. Expand \((a+b)^{3}\). 4. Expand \((a-b)^{3}\). 5. Expand \((a+b+c)^{2}\). 6. Expand \((a+b-c)^{2}\). 7. Expand \((a-b+c)^{2}\). 8. Expand \((a-b-c)^{2}\). 9. Factor \(a^{2}-b^{2}\). 10. Factor \(a^{3}-b^{3}\). 11. Factor \(a^{3}+b^{3}\). 12. Factor \(a^{4}-b^{4}\). 13. Factor \(a^{5}-b^{5}\). 14. Factor \(a^{5}+b^{5}\). 15. Factor \(a^{n}-b^{n}\). 16. Factor \(a^{n}+b^{n}\), with \(n\) odd. 17. Try to factor \(a^{2}+b^{2}\).** 18. Compute \(100003^{2}-99997^{2}\). 19. Factor \(x^{4}+y^{4}+x^{2}y^{2}\). 20. Factor \((a^{2}+9b^{2}-1)^{2}-36a^{2}b^{2}\). 21. Factor \(x^{3}+2x^{2}y+y^{3}+2xy^{2}\). 22. Factor \(a^{4}+b^{4}+c^{4}-2a^{2}b^{2}-2b^{2}c^{2}-2c^{2}a^{2}\). 23. Factor \((ax-by)^{3}+(by-cz)^{3}+(cz-ax)^{3}\). 24. Factor \(x^{8}+x^{6}+x^{4}+x^{2}+1\).25. Evaluate \[\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{2+\sqrt{3}}+\frac{1}{ \sqrt{5}+2}+\frac{1}{\sqrt{6}+\sqrt{5}}+\frac{1}{\sqrt{7}+\sqrt{6}}+\frac{1}{2 \sqrt{2}+\sqrt{7}}+\frac{1}{3+2\sqrt{2}}.\] 26. Evaluate \[\frac{1}{\sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}}+\frac{1}{\sqrt[3]{4}+\sqrt[3]{6} +\sqrt[3]{9}}+\frac{1}{\sqrt[3]{9}+\sqrt[3]{12}+\sqrt[3]{16}}+\frac{1}{\sqrt[ 3]{16}+\sqrt[3]{20}+\sqrt[3]{25}}.\] 27. If \(\frac{1}{a+b}=\frac{1}{a}+\frac{1}{b}\), find \(\left(\frac{a}{b}\right)^{3}\). 28. If \(x+\frac{1}{x}=1\), find \(x^{3}+\frac{1}{x^{3}}\). 29. If \(a+b=1\) and \(a^{2}+b^{2}=2\), compute \(a^{4}+b^{4}\). **The Problems!** 30. Factor \((x+y)(x-y)+4(y-1)\). 31. Factor \(x^{3}(x-2y)+y^{3}(2x-y)\). 32. Factor \(x^{2}y-y^{2}z+z^{2}x-x^{2}z+y^{2}x+z^{2}y-2xyz\). 33. Factor \((c^{2}+d^{2}-b^{2}-a^{2})^{2}-4(ab-cd)^{2}\). 34. Simplify the expression \[\frac{bx(a^{2}x^{2}+2a^{2}y^{2}+b^{2}y^{2})+ay(a^{2}x^{2}+2b^{2}x^{2}+b^{2}y^ {2}))}{bx+ay}.\] 35. Evaluate the sum \[\frac{1}{\sqrt{15}+\sqrt{13}}+\frac{1}{\sqrt{13}+\sqrt{11}}+\frac{1}{\sqrt{1 1}+3}+\frac{1}{3+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{5}}.\] 36. Find the four-digit numbers whose product is \(4^{8}+6^{8}+9^{8}\). 37. Find the prime factors of \(3^{18}-2^{18}\). 38. Factor \((ax+by)^{2}+(ay-bx)^{2}+c^{2}x^{2}+c^{2}y^{2}\). 39. Factor \(1+a+b+c+ab+bc+ca+abc\). 40. (HMMT 2003) Compute \[2\sqrt{\frac{3}{2}+\sqrt{2}}-\left(\frac{3}{2}+\sqrt{2}\right).\] 41. (ARML 2003) Find the largest divisor of 1001001001 that does not exceed 10000. 42. (HMMT 2005) The number 27000001 has exactly four prime factors. Find their sum. Problem Set**Theorem** (The Rational Zero Theorem).: _Let \(p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}\) be an integer polynomial with \(a_{n}\), the leading coefficient, not equal to \(0\). If the rational number \(\frac{p}{q}\) when written in lowest terms is a zero of the polynomial, then_\[p\text{ is a factor of }a_{0}\quad\text{ and }\quad q\text{ is a factor of }a_{n}.\]** 1. Factor \(63x^{2}-22x-8\). 2. Factor \(63x^{2}-3x-8\). 3. Factor \(6x^{2}-ax-15\) for (1) \(a=13\); (2) \(a=1\); (3) \(a=27\). 4. Factor \(x^{2}+\left(a+\frac{1}{a}\right)xy+y^{2}\). 5. Factor \(x^{2}+x+6y^{2}+3y+5xy\). 6. Factor \(3x^{2}-7xy-6y^{2}+7x+12y-6\). 7. Factor \(x^{4}+x^{2}y^{2}+y^{4}\). 8. Factor \(6x^{2}+xy-2y^{2}+2x-8y-8\). 9. Factor \(x^{3}-19x-30\). 10. Factor \(x^{3}+9x^{2}+26x+24\). 11. Factor \(x^{4}+2x^{3}-9x^{2}-2x+8\). 12. Factor \(2x^{4}+7x^{3}+4x^{2}-7x-6\). 13. Factor \(7x^{4}+20x^{3}+11x^{2}+40x-6\). 14. Factor \((y+1)^{4}+(y+3)^{4}-272\). 15. Factor \((a-b)^{4}+(a+b)^{4}+(a^{2}-b^{2})^{2}\). 16. Factor \((x^{2}+x+1)(x^{2}+x+2)-12\). 17. Factor \((x^{2}+3x+2)(x^{2}+7x+12)-120\). 18. Factor \(a^{3}+b^{3}+c^{3}-3abc\). 19. Factor \((x+y+z)^{3}-(x^{3}+y^{3}+z^{3})\). 20. Determine all the solutions of the system of equations \[\begin{cases}x+y+z=5 x^{3}+y^{3}+z^{3}=395\end{cases}\] in integers \((x,y,z)\).21. Find all integers \(x\) such that \(x^{4}+4\) is prime. 22. Prove that for any integer \(n\) greater than 1, the number \(n^{5}+n^{4}+1\) is composite. 23. [AIME1 2000] Suppose that \(x\), \(y\), and \(z\) are three positive numbers that satisfy the equations \[xyz=1,\quad x+\frac{1}{z}=5,\quad y+\frac{1}{x}=29.\] Then \(z+\frac{1}{y}=\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\). 24. Let \(a\), \(b\), and \(c\) be distinct nonzero real numbers such that \[a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}.\] Prove that \(|abc|=1\).**The Problems!**25. Factor \(a^{4}+64b^{4}\).26. Factor \(x^{2}-8ax-40ab-25b^{2}\).27. Factor \(x^{2}y^{2}-4xy-x^{2}-y^{2}+1\).28. Factor \((x^{2}-x-3)(x^{2}-x-5)-3\).29. Factor \(3x^{2}+5xy-2y^{2}+x+9y-4\).30. Factor \(x^{3}+6x^{2}+11x+6\).31. Factor \(4x^{3}-31x+15\).32. Factor \(6x^{4}+27x^{3}-13x^{2}+9x-5\).33. Factor \(x^{2}-xy-2y^{2}-x+5y-2\).34. Factor \((x^{2}+5x+6)(x^{2}+7x+6)-3x^{2}\).35. Find all integers \(m\) such that the number \(m^{5}+m+1\) is composite.36. A positive integer is written on each face of a cube. Each vertex is then assigned the product of the numbers written on the three faces intersecting the vertex. The sum of the numbers assigned to all the vertices is equal to 1001. Find the sum of the numbers written on the faces of the cube.37. Factor \(ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})\).38. If \(a,b,c,d,e,f,g,h,k\) are either \(1\) or \(-1\), determine the minimum value of \(aek-afh+bfg-bdk+cdh-ceg\).39. [AHSME 1999, Titu Andreescu] Determine the number of ordered pairs of integers \((m,n)\) for which \(mn\geq 0\) and\[m^{3}+n^{3}+99mn=33^{3}.\]40. Determine all the solutions of the system of equations\[\begin{cases}x+y+z=4 x^{5}+y^{5}+z^{5}=274\end{cases}\]in integers \((x,y,z)\).41. Let \(a\), \(b\), and \(c\) be distinct nonzero real numbers such that\[a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}=k.\]Find all possible values of \(k\). 1.3 Operation of Rational Expressions**Properties of fractions**1. Let \(a\), \(b\), \(c\) and \(d\) be real numbers with \(bd\neq 0\). We have the following properties: 1. for \(m\neq 0\), \(\frac{a}{b}=\frac{am}{bm}\). 2. for \(m\neq 0\), \(\frac{a}{b}=\frac{\frac{a}{m}}{\frac{b}{m}}\). 3. \(\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}\). 4. \(\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\). 5. \(\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}\). 6. if \(a\neq 0\), then \(\frac{\frac{a}{b}}{\frac{a}{b}}=\frac{b}{a}\). 7. if \(c\neq 0\), then \(\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}=\frac{ad}{bc}\). 8. if \(n\) is a positive integer, then \(\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}\). 9. if \(b+d\neq 0\) and \(\frac{a}{b}=\frac{c}{d}\), then \(\frac{a}{b}=\frac{c}{d}=\frac{a+c}{b+d}\). **2. Simplify \[\frac{2a-b-c}{a^{2}-ab-ac+bc}+\frac{2b-c-a}{b^{2}-bc-ba+ca}+\frac{2c-a-b}{c^{2 }-ca-cb+ab}.\] 3. Given real numbers \(x\), \(y\) and \(z\) such that \(xyz=1\), prove that \[\frac{1}{1+x+xy}=\frac{z}{z+zx+1}=\frac{yz}{yz+1+y}.\] 4. Which of the fractions \[\frac{5678901234}{6789012345}\quad\text{and}\quad\frac{5678901235}{6789012347}\] is greater? 5. Simplify \[\frac{b-c}{(a-b)(a-c)}+\frac{c-a}{(b-c)(b-a)}+\frac{a-b}{(c-a)(c-b)}+\frac{2} {b-a}-\frac{2}{c-a}.\]6. Find all positive integers \(n\) such that \[\frac{2}{n}+\frac{3}{n+1}+\frac{4}{n+2}=\frac{133}{60}.\] 7. Simplify \[\frac{(y-z)^{2}}{(x-y)(x-z)}+\frac{(z-x)^{2}}{(y-z)(y-x)}+\frac{(x-y)^{2}}{(z-x)( z-y)}.\] 8. (AIME 1986) What is the largest positive integer \(n\) for which \(n^{3}+100\) is divisible by \(n+10\)? 9图片描述

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