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• Algebra : You can expand an expression with normal for example (x+1)^4. Conversely, factor( or cfactor() factors an expression (like x^4-1) over Q or Q[i].
  simplify(sin(3x)/sin(x)); gcd(x^4-1,x^3-1)
• Solve equations : solve(x^2-3*x+2=0); csolve(x^2=2*i); solve([x+y=1,x-y=3],[x,y])
• Calculus : f(x):=sin(x^2):; f(sqrt(pi)); f'(2); f'(y)
  int(1/(x^4-1)); int(1/(x^4+1)^4,x,0,+infinity)
  limit(sin(x)/x,x=0); series(sin(x),x=0,5);
• Linear algebra : A:=[[1,2],[3,4]]; inv(A); det(A-x*idn(A)); A[0,0]; rref(A); eigenvalues(A); eigenvectors(A);
• Plots : plot([sin(x),x-x^3/3!],x=-3..3,color=[red,blue])
  plotfunc(x^2-y^2,[x=-2..2,y=-2..2]); plane(z=0,color=cyan+filled);