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Algebra Calculators (examples)

1. Addition, subtraction, multiplication, division of two polynomials
1.1 Addition of two polynomials eg.`(x^3-4x^2+4x-8) + (x-2)`
1.2 Subtraction of two polynomials eg.`(x^3-4x^2+4x-8) - (x-2)`
1.3 Multiplication of two polynomials eg.`(x^3-4x^2+4x-8) xx (x-2)`
1.4 Long division of two polynomials eg.`(x^3-4x^2+4x-8) -: (x-2)`
1.5 Synthetic division of two polynomials eg.`(x^3-4x^2+4x-8) -: (x-2)`
1.6 Remainder theorem eg.`(x^3-4x^2+4x-8) -: (x-2)`

2. Factoring Polynomials
2.1 Factor out GCF (Taking common) eg.`x^2+2x`
2.2 Difference of squares eg.`x^2-9`
2.3 Sum and Difference of cubes eg.`x^3-27, x^3+27`
2.4 Whole square of a binomial eg.`4x^2+12xy+9y^2`
2.5 Reverse FOIL method (Splitting the middle term) eg.`x^2+10x+24`
2.6 Perfect square trinomial eg.`4x^2+y^2+1+4xy+4x+2y`
2.7 Factorization with the help of factor theorem eg.`x^3-3x^2-6x+8`
2.8 Cyclic Expressions eg.`a^2(b-c)+b^2(c-a)+c^2(a-b)`
2.9 Factorization of fourth power polynomial eg.`x^4+x^2+1`

3. Expand and simplify polynomial
3.1 FOIL Method eg.`(2x-1)(4x+5)`
3.2 Expand Difference of Squares eg.`(x-6)(x+6)`
3.3 Expand Perfect Squares of binomial eg.`(x-3)^2`
3.4 Expand Cubes eg.`(x-3)^3`
3.5 Expand Trinomials eg.`(a+b)(a^2-ab+b^2)`
3.6 Expand Perfect Squares of trinomial eg.`(2x+3y+4z)^2`
3.7 Binomial expansion eg.`(2x+3y)^5`

4. Complete square, Is perfect square, Find missing term
1. Completing the square for quadratic equation eg. `9x^2+6x+1= 9( x+1/3 )^2`
2. Determining if the polynomial is a perfect square eg. (1) `x^2-4xy+4y^2`, (2) `3x^2+5x+2`
3. Find the missing term in a perfect square trinomial eg. (1) `9x^2` - __ + 16, (2) __ + `12x^2` + 9, (3) `49x^2` + 56 xy + __

5.1 HCF(GCD)-LCM of Polynomials eg. Find GCD, LCM of `(2x^2-4x), (3x^4-12x^2), (2x^5-2x^4-4x^3)`
5.2 Find other polynomial when one polynomial its GCD and LCM are given

6. Rational Expression of Polynomials
1. Reduce rational expressions eg. `(4x^2-25)/(8x^3-125)`
2. Adding, subtracting, multiplying, dividing of rational expressions polynomials eg. `(x-3)/(x+1)-(x-6)/(x)`

7. Simplifying Algebraic Expressions
eg. (1) `(4x+1)/(3x-2)=3/2`, (2) `(2x-1)/(3x+1)+(4x-1)/(6x)=0`

8.1 Quadratic Equation
1.1 Solving quadratic equations by factoring, eg. (1) `25x^2-30x+9=0`, (2) `x^2+10x-56=0`
1.2 Solving quadratic equations using the quadratic formula, eg. (1) `25x^2-30x+9=0`, (2) `x^2+10x-56=0`
1.3 Discriminant eg. (1) `25x^2-30x+9=0`, (2) `x^2+10x-56=0`
1.4 Discriminant & Nature of Roots eg. (1) `25x^2-30x+9=0`, (2) `x^2+10x-56=0`
2. Find the quadratic equation whose roots are `alpha` and `beta` eg. (1) `alpha=3, beta=-4`, (2) `alpha=1+3sqrt(2), beta=1-3sqrt(2)`
3. Roots for non-zero denominator eg. (1) `(5x-18)/(x+2)=(2x-6)/(x-1)`, (2) `(x)/(x+1)+(x+1)/(x)=5/2`, (3) `4((4x+1)/(4x-1))^(2)+(4x+1)/(4x-1)=3`, (4) `(4x+1)/(4x-1)+(4x-1)/(4x+1)=3`
4. Roots of non-quadratic equation eg. (1) `6(x^2+1/x^2)-25(x-1/x)+12=0`, (2) `(x^2+1/x^2)-8(x+1/x)+14=0`

5. If `alpha` and `beta` are roots of equation `2x^2-3x-6=0`, then find `alpha^2+beta^2`
6. If `alpha` and `beta` are roots of equation `2x^2-3x-6=0`, then find equation whose roots are `alpha^2` and `beta^2`
7. Find value of `k` for which `2x^2+kx+2=0` has real roots

9. Solve linear equation in two variables by (eg. Solve `7y+2x-11=0` and `3x-y-5=0` using Substitution method)
1. Substitution method
2. Elimination method
3. Cross multiplication method
4. Addition-Subtraction method
5. Inverse matrix method
6. Cramer's Rule method
7. Graphical method

10. Solve linear equation of any number of variables (simultaneous equations) using
1. Inverse Matrix method
2. Cramer's Rule method
3. Gauss-Jordan Elimination method
4. Gauss Elimination Back Substitution method
5. Gauss Seidel method
6. Gauss Jacobi method
7. Elimination method
8. LU decomposition method / Crout's method
9. Cholesky decomposition method
10. SOR (Successive over-relaxation) method
11. Relaxation method

11.1 Find the value of h,k
1. Find the value of h,k for which the system of equations has a Unique solution
2. Find the value of h,k for which the system of equations has Infinite solution
3. Find the value of h,k for which the system of equations has No solution
4. Find the value of h,k for which the system of equations is consistent
5. Find the value of h,k for which the system of equations is inconsistent

11.2 Determine whether the system of linear equations
1. Determine whether the system of linear equations has a Unique solution
2. Determine whether the system of linear equations has Infinite solution
3. Determine whether the system of linear equations has No solution
4. Determine whether the system of linear equations is consistent
5. Determine whether the system of linear equations is inconsistent

12. Variation Equations
1. Find value of variation using given value
(1) x `prop` y and x=6 when y=3. Find y=? when x=18, (2) x `prop` `y/z`. x=8 when y=4 and z=3. Find x=? when y=6 and z=4.
2. Prove results for given variation
(1) If x `prop` y then prove that `x^3+y^3 prop x^2y-xy^2`, (2) If 3x-5y `prop` 5x+6y then prove that x `prop` y.

13. If `x+1/x=2` then find `x-1/x, x^2-1/x^2, x^3+1/x^3`
1. If `x-1/x=6` then find (1) `x^2+1/x^2` (2) `x+1/x` (3) `x^2-1/x^2`
2. If `x+y=5` and `xy=6` then find `x^2+y^2`
3. If `x^2+y^2+z^2=29` and `xy+yz+zx=-14` then find `x+y+z`
4. If `x+y+z=1,xy+yz+zx=-1` and `xyz=-1` then find `x^3+y^3+z^3`

14. Interval notation and set builder notation eg. (1) `3 <= x <=7`, x is odd. (2) `|x^3-2| <= 25`, x in Z. (3) `x in[2,8)`

15. Set Theory eg. `A={x<=5; x in N}, B={2<=x<=8; x in N}, C={x^3-3x^2-4x=0},` Find
1. Union eg. `A uu (B uu C)=(A uu B) uu C`
2. Intersection eg. `A nn (B uu C)=(A nn B) uu (A nn C)`
3. Complement eg. `(A uu B)'=A' nn B'`
4. Power set (Proper Subset) eg. `P(A)`
5. Difference eg. (1)`A-B`, (2) `A-(B uu C)=(A-B) nn (A-C)`
6. Symmetric difference eg. (1)`A Delta B`, (2) `B Delta C`, (3) `A Delta C`
7. Cross Product eg. `A xx B`
8. Prove that any two expression is equal or not eg. `A-(B uu C)=(A-B) nn (A-C)`
9. Cardinality of a set eg. `n(A)`
10. is Belongs to a set eg. `2inB` ?
11. is Subset of a set eg. `AsubB` ?
12. is two set Equal or not eg. `A=B` ?

16. Functions
1. Find Range of `f:A->B` eg. 1. `f(x)=5x+2` where `A={1<=x<5}`, 2. `f(x)=sqrt(x)` where `A={1,4,16,36}`
2. Composite functions and Evaluating functions
eg. 1. `f(x)=2x+1`, `g(x)=x+5`. Find `fog(x)`, also evaluate at `x=2`
2. `fog(x)=(x+2)/(3x), f(x)=x-2`. Find g(2).
3. `gof(x)=1/x^2, f(x)=2+x^2`. Find g(x).
3. Find value eg. 1. `f(x)=x(x+1)(2x+1)`. Find `f(x)-f(x-1)`, 2. `f(x)=x^2-2^x`. Find `f(2)-f(0)`
4. Verifying if two functions are inverses of each other eg. 1. `f(x)=x+3,g(x)=x-3`, 2. `f(x)=4x-3,g(x)=(x+3)/4`, 3. `f(x)=x/(x-1),g(x)=(2x)/(2x-1)`

17. Functions
1. Domain of a function
2. Range of a function
3. Inverse of a function
4. Properties of a function
5. Parabola Vertex form
6. Parabola Focus
7. axis symmetry of a parabola
8. Parabola Directrix
9. Intercept of a function
10. Parity of a function
11. Asymptotes of a function

18. Descartes' rule of signs eg. `x^5-x^4+3x^3+9x^2-x+5`

19. Ratio and Proportion
1. If `a:b:c=2:3:5` then find value of `(a^2+b^2+c^2)/(ab+bc+ca)`
2. If `a:b=2:3,b:c=4:5` then find `a:b:c`
3. If `a/b=c/d=e/f` then prove that `(2a+3c-4e)/(2b+3d-4f)=(5a-4c+3e)/(5b-4d+3f)`
4. If `x/(y+z)=y/(z+x)=z/(x+y)` then prove the value of each ratio is `1/2` or `-1`
5. Geometric Mean
6. Ratios(duplicate, triplicate) and proportional(mean, third, fourth)
6.1 Duplicate Ratio
6.2 Triplicate Ratio
6.3 Sub-Duplicate ratio
6.4 Sub-Triplicate Ratio
6.5 Compounded Ratio
6.6 Mean proportional
6.7 Third proportional
6.8 Fourth proportional
6.9 Compare ratios

20. Partial Fraction decomposition eg. `(5x-4)/(x^2-x-2)`

21. Logarithmic equations eg. `log(20)+log(30)-1/2log(36)`

22. Simple Interest
23. Compound Interest
24. Percentage

25. Arithmetic Progression
26. Geometric Progression

27. Present value, Future value
27.1 Future value using Simple Interest
27.2 Future value using Compound Interest
27.3 Future value of Annuity
27.4 Future value of Annuity Due
27.5 Present value using Simple Interest
27.6 Present value using Compound Interest
27.7 Present value of Annuity
27.8 Present value of Annuity Due
27.9 Contineous Compounding

28.1 Future value factor
28.2 Present value factor
28.3 Discount factor
28.4 Future value factor table
28.5 Present value factor table
28.6 Discount factor table

29.1 Effective interest rate
29.2 Nominal interest rate

30. Amortization

31. Net Present Value (NPV)

32. Polynomial
1. Polynomial in ascending order
2. Polynomial in descending order
3. Degree of a polynomial
4. Leading term of a polynomial
5. Leading coefficient of a polynomial
6. Determine expression is a polynomial or not
7. Zeros of a polynomial

33.1 Sine Calculator
33.2 Cosine Calculator
33.3 Tangent Calculator
33.4 Cosecant Calculator
33.5 Secant Calculator
33.6 Cotangent Calculator

34.1 Inverse Sine Calculator
34.2 Inverse Cosine Calculator
34.3 Inverse Tangent Calculator
34.4 Inverse Cosecant Calculator
34.5 Inverse Secant Calculator
34.6 Inverse Cotangent Calculator

35.1 Hyperbolic Sine Calculator
35.2 Hyperbolic Cosine Calculator
35.3 Hyperbolic Tangent Calculator
35.4 Hyperbolic Cosecant Calculator
35.5 Hyperbolic Secant Calculator
35.6 Hyperbolic Cotangent Calculator

36.1 Inverse Hyperbolic Sine Calculator
36.2 Inverse Hyperbolic Cosine Calculator
36.3 Inverse Hyperbolic Tangent Calculator
36.4 Inverse Hyperbolic Cosecant Calculator
36.5 Inverse Hyperbolic Secant Calculator
36.6 Inverse Hyperbolic Cotangent Calculator

37. Law of Sines Calculator
38. Law of Cosines Calculator

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