Some fundamental algebraic formulas and identities Part 1

 

Algebra is a broad field of mathematics that encompasses various concepts and formulas. Here are some fundamental algebraic formulas and identities:

1. Basic Arithmetic Formulas:

• Addition: $�+�$
• Subtraction: $�-�$
• Multiplication: $�\times �$
• Division: $\frac{�}{�}$

2. Linear Equations:

• $��+�=0$ (For solving linear equations)

3. Quadratic Equations:

• The quadratic formula for $�{�}^2+��+�=0$: $�=\frac{-�\pm \sqrt{{�}^2-4��}}{2�}$

4. Exponents:

• ${�}^{�}$ (a raised to the power of n)

5. Logarithms:

• ${\log }_{�}(�)$ (Logarithm of x to the base b)

6. Polynomials:

• ${�}_{�}{�}^{�}+{�}_{�-1}{�}^{�-1}+\dots +{�}_1�+{�}_0$ (General form of a polynomial)

7. Factoring Formulas:

• ${�}^2-{�}^2=(�+�)(�-�)$ (Difference of squares)
• ${�}^3-{�}^3=(�-�)({�}^2+��+{�}^2)$ (Difference of cubes)
• ${�}^3+{�}^3=(�+�)({�}^2-��+{�}^2)$ (Sum of cubes)

8. Binomial Theorem:

• $(�+�{)}^{�}={\sum }_{�=0}^{�}\left(\genfrac{}{}{0px}{}{�}{�}\right){�}^{�-�}{�}^{�}$ (Expanding a binomial raised to a positive integer power)

9. Arithmetic Series:

• The sum of an arithmetic series: ${�}_{�}=\frac{�}{2}({�}_1+{�}_{�})$

10. Geometric Series:

• The sum of a geometric series: ${�}_{�}=\frac{{�}_1(1-{�}^{�})}{1-�}$

11. Pythagorean Theorem:

• In a right-angled triangle, ${�}^2+{�}^2={�}^2$ (where c is the hypotenuse)

12. Complex Numbers:

• ${�}^2=-1$ (where $�$ is the imaginary unit)
• Complex conjugate of $�+��$ is $�-��$

13. Inequalities:

• $�>�$: a is greater than b
• $�<�$: a is less than b
• $�\ge �$: a is greater than or equal to b
• $�\le �$: a is less than or equal to b

14. Arithmetic Mean (Average):

• The arithmetic mean of $�$ numbers ${�}_1,{�}_2,\dots ,{�}_{�}$ is given by: $\text{Mean}=\frac{{�}_1+{�}_2+\dots +{�}_{�}}{�}$

15. Quadratic Equation (Vertex Form):

• The vertex form of a quadratic equation $�{�}^2+��+�$ is: $�=�(�-ℎ{)}^2+�$ where $(ℎ,�)$ is the vertex.

16. Distance Formula:

• The distance $�$ between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ in a coordinate plane is given by: $�=\sqrt{({�}_2-{�}_1{)}^2+({�}_2-{�}_1{)}^2}$

17. Midpoint Formula:

• The midpoint $�$ between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ is given by: $�(\frac{{�}_1+{�}_2}{2},\frac{{�}_1+{�}_2}{2})$

18. Systems of Linear Equations (Two Variables):

• For a system of equations $��+��=�$ and $��+��=�$, the solution is given by: $�=\frac{��-��}{��-��},�=\frac{��-��}{��-��}$

19. Laws of Exponents:

• ${�}^{�}\cdot {�}^{�}={�}^{�+�}$
• $\frac{{�}^{�}}{{�}^{�}}={�}^{�-�}$
• $({�}^{�}{)}^{�}={�}^{��}$
• ${�}^0=1$ (for $�\mathrm{\ne }0$)

20. Matrix Multiplication:

• If $�$ is an $�\times �$ matrix and $�$ is an $�\times �$ matrix, then the product $�=��$ is an $�\times �$ matrix. The element ${�}_{��}$ of $�$ is given by: ${�}_{��}={\sum }_{�=1}^{�}{�}_{��}{�}_{��}$

21. Discriminant of a Quadratic Equation:

• For a quadratic equation $�{�}^2+��+�=0$, the discriminant is given by: $\mathrm{\Delta }={�}^2-4��$
  • If $\mathrm{\Delta }>0$, two distinct real solutions.
  • If $\mathrm{\Delta }=0$, one real solution (repeated).
  • If $\mathrm{\Delta }<0$, two complex conjugate solutions.

22. Permutations and Combinations:

• Permutations of $�$ distinct objects taken $�$ at a time: ${}^{�}{�}_{�}=\frac{�!}{(�-�)!}$
• Combinations of $�$ distinct objects taken $�$ at a time: ${}^{�}{�}_{�}=\left(\genfrac{}{}{0px}{}{�}{�}\right)=\frac{�!}{�!(�-�)!}$

23. Fundamental Theorem of Algebra:

• Every non-constant polynomial has at least one complex root.

24. Arithmetic Sequence:

• The $�$-th term (${�}_{�}$) of an arithmetic sequence with first term ${�}_1$ and common difference $�$ is given by: ${�}_{�}={�}_1+(�-1)�$

25. Geometric Sequence:

• The $�$-th term (${�}_{�}$) of a geometric sequence with first term ${�}_1$ and common ratio $�$ is given by: ${�}_{�}={�}_1\cdot {�}^{(�-1)}$

26. Binomial Coefficient Identity:

• The identity for binomial coefficients is given by: $\left(\genfrac{}{}{0px}{}{�}{�}\right)+\left(\genfrac{}{}{0px}{}{�}{�+1}\right)=\left(\genfrac{}{}{0px}{}{�+1}{�+1}\right)$

27. Sum of the First $�$ Natural Numbers:

• The sum of the first $�$ natural numbers is given by: ${�}_{�}=\frac{�(�+1)}{2}$

28. Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality):

• For any non-negative real numbers ${�}_1,{�}_2,\dots ,{�}_{�}$, the inequality is: $\frac{{�}_1+{�}_2+\dots +{�}_{�}}{�}\ge \sqrt[�]{{�}_1\cdot {�}_2\cdot \dots \cdot {�}_{�}}$

29. Viète's Formulas:

• For a quadratic equation $�{�}^2+��+�=0$, the sum of roots ${�}_1$ and ${�}_2$ and the product of roots is given by: ${�}_1+{�}_2=-\frac{�}{�},{�}_1\cdot {�}_2=\frac{�}{�}$

30. De Moivre's Theorem:

• For any real number $�$ and integer $�$, $(\cos �+�\sin �{)}^{�}=\cos (��)+�\sin (��)$

31. Law of Cosines:

• In a triangle with sides $�$, $�$, and $�$, and angles $�$, $�$, and $�$, the Law of Cosines is: ${�}^2={�}^2+{�}^2-2��\cos �$

32. Law of Sines:

• In a triangle with sides $�$, $�$, and $�$, and angles $�$, $�$, and $�$, the Law of Sines is: $\frac{�}{\sin �}=\frac{�}{\sin �}=\frac{�}{\sin �}$

33. Euler's Formula:

• Euler's formula relates complex exponentials to trigonometric functions: ${�}^{��}=\cos �+�\sin �$

34. Wilson's Theorem:

• For a prime number $�$, $(�-1)!\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}�)$

35. Principal Square Root:

• The principal square root of a non-negative real number $�$ is denoted by $\sqrt{�}$, and $\sqrt{�}\cdot \sqrt{�}=�$

36. Partial Fraction Decomposition:

• For a rational function, the process of expressing it as the sum of simpler fractions is known as partial fraction decomposition.

37. Cramer's Rule:

• Cramer's Rule is a method for solving a system of linear equations using determinants. For a system $��=�$, if the determinant of the coefficient matrix $\mathrm{\mid }�\mathrm{\mid }$ is non-zero, the solution is given by: ${�}_{�}=\frac{\mathrm{\mid }{�}_{�}\mathrm{\mid }}{\mathrm{\mid }�\mathrm{\mid }}$ where ${�}_{�}$ is the matrix obtained by replacing the $�$-th column of $�$ with vector $�$.

38. Inverse Trigonometric Identities:

• ${\sin }^{-1}(�)+{\cos }^{-1}(�)=\frac{�}{2}$
• ${\tan }^{-1}(�)+{\cot }^{-1}(�)=\frac{�}{2}$
• ${\sec }^{-1}(�)+{\csc }^{-1}(�)=\frac{�}{2}$

39. Pascal's Identity:

• Pascal's Identity states that $\left(\genfrac{}{}{0px}{}{�}{�-1}\right)+\left(\genfrac{}{}{0px}{}{�}{�}\right)=\left(\genfrac{}{}{0px}{}{�+1}{�}\right)$

40. Distance between Point and Line:

• The distance $�$ between a point $({�}_0,{�}_0)$ and a line $��+��+�=0$ is given by: $�=\frac{\mathrm{\mid }�{�}_0+�{�}_0+�\mathrm{\mid }}{\sqrt{{�}^2+{�}^2}}$

41. Sum of Cubes:

• ${�}^3+{�}^3=(�+�)({�}^2-��+{�}^2)$

42. Bayes' Theorem:

• Bayes' Theorem relates conditional and marginal probabilities: $�(�\mathrm{\mid }�)=\frac{�(�\mathrm{\mid }�)\cdot �(�)}{�(�)}$

43. Heron's Formula:

• Heron's formula gives the area ($�$) of a triangle with sides $�$, $�$, and $�$: $�=\sqrt{�(�-�)(�-�)(�-�)}$ where $�$ is the semi-perimeter of the triangle, $�=\frac{�+�+�}{2}$.

44. Completing the Square:

• Completing the square is a method used to solve quadratic equations by expressing them in the form $(�-ℎ{)}^2=�$.

45. Euler's Totient Function:

• Euler's Totient Function $�(�)$ gives the count of positive integers less than $�$ that are coprime to $�$.

46. Laplace Transform:

• The Laplace transform of a function $�(�)$ is given by $\mathcal{�}\{�(�)\}=�(�)$, where $�$ is a complex number.

47. Binomial Theorem (General Term):

• The general term of the binomial expansion of $(�+�{)}^{�}$ is given by: $\left(\genfrac{}{}{0px}{}{�}{�}\right){�}^{�-�}{�}^{�}$

48. Mobius Inversion Formula:

• The Möbius inversion formula relates the summation of arithmetic functions: $�(�)={\sum }_{�\mathrm{\mid }�}�(�)⟺�(�)={\sum }_{�\mathrm{\mid }�}�(�)�\left(\frac{�}{�}\right)$ where $�$ is the Möbius function.

49. Sum of Arithmetic Series:

• The sum ${�}_{�}$ of an arithmetic series with $�$ terms, first term ${�}_1$, and common difference $�$ is given by: ${�}_{�}=\frac{�}{2}(2{�}_1+(�-1)�)$

50. Sum of Geometric Series:

• The sum ${�}_{�}$ of a geometric series with $�$ terms, first term ${�}_1$, and common ratio $�$ is given by: ${�}_{�}=\frac{{�}_1\cdot (1-{�}^{�})}{1-�}$