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 Algebra Calculators 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) 2. Factoring Polynomials eg. (1) 25x^2-36, (2) 4x^2+12xy+9y^2, (3) x^3-3x^2-6x+8, (4) a^2(b-c)+b^2(c-a)+c^2(a-b) 3. Expand and simplify polynomial eg. (1) (X+2)(X+3), (2) (2X+3Y)^2, (3) (2X+3Y+4Z)^2, (4) 102^2, (5) 102 xx 106 (Expand using the identity) 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. HCF(GCD)-LCM of Polynomials eg. Find GCD, LCM of (2X^2-4X), (3X^4-12X^2), (2X^5-2X^4-4X^3) 6. Rational Expression of Polynomials 1. Reduced terms of 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. 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 & 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 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 11. 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. 12. 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 13. 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) 14. 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 ? 15. 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. f(x)=2x+1, g(x)=x+5. Find fog(x), gof(x), fog(2), f(-3), g(4) 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) 16. Descartes' rule of signs eg. x^5-x^4+3x^3+9x^2-x+5 17. 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