Polynomials & Scientific Calculator
(Last update: 2023/09/13 -- v10.0.26)

Welcome here to this Free Online Polynomial, and Scientific Calculator. You can perform operations with polynomials (+, -, *, /), find roots; operate logarithmic, trigonometric, matrix or exponential functions, or define custom functions, in the field of real or complex numbers. The calculator incorporates the constants 'pi' and 'e'.

You may indicate decimal binary, octal, deciaml or hexadecimal numbers employing &b, &o, &d and &h. For example input '&h 0f &d' (without quotes) will output decimal 15. Other options are the calculus of derivatives, integrals, Lagrangian interpolation, matrices operations and resolution of systems of equations. In many cases, you may glance at the examples or to the intermediate steps.

Save:
Functions:
---
Vectorial
---------
CROSS
DOT
POLY
Numeric
-------
ABS
EXP
FLOOR
LN
LOG
LOGTEN
LOGTWO
ROUND
SIGN
SQR
Complex
-------
ARG
CONJ
IM
NORM
RE
Trigonometric
-------------
ACOS
ACOSH
ACOT
ACOTH
ACSC
ACSCH
ASEC
ASECH
ASIN
ASINH
ATAN
ATANH
COS
COSH
COT
COTH
CSC
CSCH
SEC
SECH
SIN
SINH
TAN
TANH
Polynomial
----------
FACTOR
GCD
LAGRANGIANINTERPOLATION
LCM
MOD
ORTHOG
RESIDUE
ROOTS
Other
-----
EULERS
ISOLATE
LIM
Matrix
------
ADJ
COF
DET
ECHELON
EGV
EGVL
EIGENVALUES
EIGENVECTORS
IDENTITY
JACOBIAN
JORDAN
RANK
TRACE
TRANSPOSE
TRN
Examples:
---Basic operations:---
2×3+4
2^3^2
√(2+2)
1600/((1-(1+1/4)^30)/(1-(1+1/4)))
m!/(n!*(m-n)!)@m=5@n=3
gcd(12;36;6;24) ' Greatest common divisor
lcm(12;18;60) ' Least common multiple
---Custom functions:---
P(x)/Q(x)?P(x)=(x+1)*(x-1)|Q(x)=x+1
y^2?y=(x+i)*(x-i)
f(3,1)?f(x,y)=y-x
f(2,-2)?f(x,y)=y-x;x+y|x*y;-2*x*y
---Evaluating:---
x^2-9*x+20@x=5
x^2-9x+20@x=4.5
2+x/(y*z)@x=12|y=4|z=-3
---Hexadecimal, octal, binary, decimal:
&h0F+&o17+&b1111+&d15
---Logical operators:
(&h0ff OR &h0f) AND &b111
---Modulus:
((x5+3x3+4)*(6x6+4x3))%11
(x7+x6+x5+x4+x3+x2)mod(x5+x)
---POLYNOMIALS:
---Find polynomial given zeros:
POLY(1;i;-i) ' find polynomial of given roots 1, i and -i
POLY(2;5) ' find polynomial of given roots 2 and 5
---Add, subs., multiply, divide:
(x-1)+(x+1)
(x-1)-(x+1)
(x-1)*(x+1)
(x^2-1)/(x+1)
gcd((3x2-3)(x+2);-(3x-3)(x+2);(2x+4))' Greatest common divisor
poly(-1;2) ' find a polynomial given the roots
---Root finder:
roots(x16-1)
factors(x16-1)
(x-1)*(x+1)=0
(5*x+2)*(3*x-1)=0
x^2-9x+20=0
---Partial fraction descomposition:
partialfractions((-4s+8)/(s2+6s+8))
partialfractions((s-2)/((s-1)(s+2)(s-3)))
---Orthogonality:
orthog(-1;1;2*t;3*t^2+5)
---Interpolation:---
lagrangianinterpolation(-2;1|0;-1|2;1)
lagrangianinterpolation(-1;1;10;abs(x))' 10 intervals at [-1,1] with Chebyshev nodes for f(x)=abs(x)
---Derivatives:---
Dx(cosh(x))
Dy(sin(x^2*y^2+y))
Dy(z)?z=x+y|x=y^2
---2nd.derivative:---
D2x(3x^2-2x+5)
---Jacobian:---
jacobian(1,000x^2-2x+5;sin(x2)|xsinh(y);xasinh(y))
---Vector operation:
dot(3;-3;1|4;9;2)
cross(3;-3;1|4;9;2)
---LIMITS:
lim((x2-16)/(x-4);x;4)
lim(sin(x)/x;x;0)
lim(ln(x)/x;x;0)
lim((5t4-4t2-1)/(10-t-9t3);t;1)
lim(x*e^x;x;-infinity)
lim(exp(x)/x^2;x;∞)
lim(x*ln(x);x;0)
lim(x^(1/x);x;infinity)
---MATRIX operations:
(1;0|0;-1|^|-1)|*|1;0|0;-1
(1;2|3;4|+|1;0|0;1)|*|-1;0|0;1
((1;0|0;3|-|5;0|3;-2)|^|-1)|*|(-4;0|-3;5)
(x;0|0;1|^|-1)|*|a?a=x;0|0;1
x+1;0|0;-1|^|-1|*|a?a=(x+1);0|0;-1
A+(-B*C)?A=1;2|3;-1||B=A|C=1;0|0;1
Dx(x^2-sin(x);-6*x^2-x+3|tan(x)+x^2;sinh(x))
integral(A)dy @ A = x^2*y-y^2;-6*x*y^2-x+3|1/(y+1);sinh(y)
---Inverse matrix:
A^-1?A=1;2|3;-1
A^-1*A?A=1;2|3;-1
A^-1*A?A=x;y|2;-3
a*a^-1|*|a|-|a?a=cos(x);-sin(x)|sin(x);cos(x)
A*A^-1?A=(3;a;a;a;a|a;3;a;a;a|a;a;3;a;a|a;a;a;3;a|a;a;a;a;3)
---Determinant:
det(A)?A=(1-x;2|3;2-x)
roots(det(A))?A=(1-x;2|3;2-x)
det(A)=0?A=(1-x;2|3;2-x)
roots(det(A))@A=(3;a;a;a;a|a;3;a;a;a|a;a;3;a;a|a;a;a;3;a|a;a;a;a;3)
det(A)|-|(4a+3)*(3-a)^4?A=(3;a;a;a;a|a;3;a;a;a|a;a;3;a;a|a;a;a;3;a|a;a;a;a;3)
det(A-B)=0?A=(1;2|3;2)|B=λ*(1;0|0;1)
---Identity matrix:
Identity(3) ' 3x3 Identity matrix
Identity(3;4) ' 3x4 Identity matrix
---Trace:
trace(1-x;2|3;2-x)
---Transpose:
transpose(1-x;2|3;2-x)
---Cofactor:
cof(1;3;1|1;1;2|2;3;4)
---Adjoint:
Adj(1;3;1|1;1;2|2;3;4)
---Echelon:
Echelon(1;2;1|-2;-3;1|3;5;0) /* see https://en.wikipedia.org/wiki/Row_echelon_form */
---Rank:
Rank(1;2;1|-2;-3;1|3;5;0) /* see https://en.wikipedia.org/wiki/Rank_(linear_algebra) */
---EigenValues:
eigenvalues(2;1|1;2)
---Eigenvectors:
eigenvectors(2;1|1;2)
---Jordan form:
A*T-T*jordan(A)?A=2;0;1|0;2;0|1;0;2|T=eigenvectors(A)
---SYSTEM OF EQUATIONS:---
---Linear:---
&var1 Xc=1/(2*pi*f*C)?Xc=50|f=10e6
x+y=2|x-y=1
---Non-Linear:---
x2-10x+y2+8=0|x*y2+x-10y+8=0
---INTEGRATION:
---Integration of polynomials:
integral(6x-2)
integral(3x2+y)dx
∫(x2+y)dy
∫(√x+(1/√x))^2|∫((√x+(1/√x))^2)dx|(√x+(1/√x))^2
---Inmediate integrals:
∫(sin(x))dx
∫(ln(x))dx
---Trigonometric integrals:
∫(sin2(x)*cos(x))dx
∫(tan2(x))dx
---Definite integrals:
integral(1,2,x)dx
integral(1,2,x*y)dy
∫(1,2,x)dx
∫(a,b,x*y)dx?a=1|b=2
∫(a,b,S(x))dx,pi(10^(3/2)-1)/27?a=0|b=1|f(x)=x3|S(x)=2*pi*f(x)*sqr(1+(Dx(f(x)))^2)
---Partial fractions integration:
∫[1/(x(x-2)^2)]dx
(x(x-2)^2)*Dx(∫1/(x(x-2)^2)dx)
---Trigonometric integration:
∫(sec(x)*tan2(x))dx
---Integration by parts:
13*∫(sin(3*x)*exp(2*x))dx
---Improper integrals
lim(integral(0;b;1/(1+x2))dx,b,infinity)
---DIFFERENTIAL EQUATIONS:
---First order linear:
Dt(y)=9.8-0.196y@y(0)=48
t*Dt(y)+2*y=t^2-t+1@y(1)=0.5
2*Dt(y)-y=4*sin(3t)@y(0)=_y0
---First order separable:
Dx(y)=6y^2x@y(1)=1/25
Dx(y)=(3x2+4x-4)/(2y-4)@y(1)=3
Dx(y)=exp(-y)*(2x-4)@y(5)=0
1/(exp(-y)*(2x-4))*Dx(y)=1@y(5)=0
Dθ(r)=r^2/θ@r(1)=2
Dt(y)=exp(y-t)*sec(y)*(1+t^2)@y(0)=0
---First order exact:
2xy-9x2+(2y+x2+1)*Dx(y)=0@y(0)=-3
2xy2+4=2*(3-x2y)*Dx(y)@y(-1)=8
(2xy2+4)/(2*(3-x2y)*Dx(y))=1@ y(-1)=8
---First order Bernoulli:
Dx(y)+(4/x)*y = x3*y2@y(2)=-1
Dx(y)=5y+exp(-2x)*y^-2@y(0)=2
6*Dx(y)-2y = xy4 @ y(0)=-2
Dx(y)+y/x-sqr(y)=0 @ y(1)=0
---First order substitution:
xyDx(y)+4x2+y2=0 @ y(2)=-7
Dx(y)=exp(9y-x) @ y(0)=0
---Euler's method:
Eulers(_step;N;Dt(y)-y=-1/2*exp(t/2)*sin(5t)+5*exp(t/2)*cos(5t))?_step=0.1|N=10|y(0)=0
---MISCELLANEOUS:---
isolate(C,w*L=1/(w*C))
isolate(x,x-y+3=-1)
Definitions box (e.g. P(x)=x^2-1, x=3, ...):