(3) Algebraic ComputationsMaple is a very powerful algebraic calculator that can manipulate and find solutions to many problems symbolically. You can use variables instead of specifying numbers when defining a problem. This capability allows you to ask "What if...?" types of questions.

Maple provides different ways of manipulating and displaying expressions to make them easier to verify, or more effective to use. This flexibility allows you to do such things as expand binomials, factor results, simplify trigonometric expressions, assign results to variable names, and convert expressions to different forms.

Maple can expand binomials such as NiMqJCwmJSJ4RyIiIiUieUdGJiIjOg==. The following Maple commands create the expression, and then expand it.expr := (x+y)^15;expand( expr );After viewing the results, use the factor command to factor the last result and check the computation.factor( % );Asigna NiMqJCwmKiRJInhHNiIiIiMiIiIqJiIiJEYpSSJ5R0YnRikhIiIiIio= al nombre miexpress y luego expande la expresi\363n.Factoriza el resultado anteriorFor more information on the expand and factor commands, see ?expand and ?factor in the Maple help system.

Maple can apply identities to simplify many lengthy mathematical expressions, such as trigonometric expressions. Consider NiMsLCokLSUkY29zRzYjJSJ4RyIiJiIiIiokLSUkc2luR0YnIiIlRioqJiIiI0YqKiRGJUYwRipGKiomRjBGKiokRixGMEYqISIiLUYmNiMqJkYwRipGKEYqRjQ=. The following Maple commands simplify the expressionsimplify( cos(x)^5 + sin(x)^4 + 2*cos(x)^2 - 2*sin(x)^2 - cos(2*x) );\277Cual es el resultado esperado al simplificar la expresi\363n NiMsJi1JJXNxcnRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYjLCgqJC1JJHNpbkdGJjYjSSJ4R0YpIiIjIiIiKiQtSSRjb3NHRiZGL0YxRjIiIilGMkYyKiYiIiZGMiokKiRGMCIiJEY7RjIhIiI= ? SimplificalaAnother way to simplify expressions is to use the normal command, which puts fractions on a common denominator and removes common factors in the numerator and denominator. The fraction NiMqJiwmKiQlInhHIiIkIiIiKiQlInlHRichIiJGKCwqKiRGJiIiI0YoRiZGKEYqRisqJEYqRi5GK0Yr is much simpler after Maple removes the common factors with the comand normal( (x^3-y^3)/(x^2+x-y-y^2) );\277Cual es el resultado esperado al normalizar la expresi\363n NiMqJiwmKiQtSSJmRzYiNiNJInhHRigiIiMiIiJGLCEiIkYsLCZGJkYsRixGLUYt ?. HazloNormaliza la expresi\363n NiM8Iy8sJiomIiIjIiIiSSJ4RzYiISIiRigqJkkieUdGKkYoIiIkRitGKCIiIQ==Normaliza las expresiones NiM8JComLCYqJC1JImZHNiI2I0kieEdGKSIiIyIiIkYtISIiRi0sJkYnRi1GLUYuRi4vLCYqJkYsRi1GK0YuRi0qJkkieUdGKUYtIiIkRi5GLSIiIQ== y NiM3JComLCYqJC1JImZHNiI2I0kieEdGKSIiIyIiIkYtISIiRi0sJkYnRi1GLUYuRi4vLCYqJkYsRi1GK0YuRi0qJkkieUdGKUYtIiIkRi5GLSIiIQ== separadamente. \277Cual es la diferencia entre las dos respuestas anteriores? Ver Sets and Lists (Conjuntos y Listas)For more information on the simplify and normal commands, see ?simplify and ?normal in the help system.

As mentioned in (1) Working through the New User's Tour, you can assign the results of a computation to a Maple name (sometimes referred to as a variable). Assigning to names is essential for managing large numbers of expressions and results, especially if you want to reuse them later in a session. For example, create the expression NiMqJiwoKiYiI1QiIiIqJCUieEciIiNGJ0YnRilGJ0YnRidGKiwmKiZGKkYnRilGJ0YnRichIiJGJw==, and store it as NiMlJmV4cHIxRw==.Use the expand command on NiMlJmV4cHIxRw== and store the result as a variable, NiMlJmV4cHIyRw==.Evaluate NiMlJmV4cHIyRw== at x=1. ( use eval( expr2 , x=1 ))Asigna los nombres numerador1 y denominador1 a las expresiones expr2 (del ejercicio anterior) y a la expansi\363n de NiMqJiwmKiYiIiQiIiJJInhHNiJGJ0YnIiImRidGJywmKiYiIiNGJ0YoRidGJ0YnISIiRic= , respectivamente. Asigna el nombre CocNorm a la normalizaci\363n del cociente cuyo numerador y denominador son numerador1 y denominador1 respectivamente. Finalmente eval\372e CocNorm en x=3 y en x=-5/3. For more information on the normal command, see ?normal in the help system.

The convert command allows you to convert many types of expressions into specific forms. For a complete list of conversions available in Maple, see ?convert.The following example converts a symbolic expression, NiMqJiwmKiYlImFHIiIiKiQlInhHIiIjRidGJyUiYkdGJ0YnKiZGKUYnLCgqJiIiJEYnRihGJyEiIkYpRjAiIiVGJ0YnRjA= , into its partial fraction decomposition.my_expr := (a*x^2+b)/(x*(-3*x^2-x+4));convert( my_expr, parfrac, x );The following example converts a trigonometric expression, NiMtJSRjb3RHNiMlInhH, into an exponential expression. See ?convert/exp convert( cot(x), exp );Encuentre la descomposici\363n en suma de fracciones simples de las expresiones racionales NiMqJiwqKiYiIiMiIiIqJEkieEc2IiIiJUYnRicqJEYpIiIkRidGKSEiIkYnRidGJywuKiRGKSIiJkYnRihGLiomRiZGJ0YsRidGLiomRjFGJyokRilGJkYnRicqJkYxRidGKUYnRi5GJkYnRi4= y NiMqJiwuKiRJInhHNiIiIiYiIiIqJEYmIiIlISIiKiYiIiNGKSokRiYiIiRGKUYsKiZGKEYpKiRGJkYuRilGKSomRihGKUYmRilGLEYuRilGKSwqKiZGLkYpRipGKUYpRi9GKUYmRixGKUYpRiw= Convierta 35 en base binaria ( use convert(35, binary))Convierta la forma decimal peri\363dica 1.23456 en su forma racional ( use convert( 1.23456, fraction ))For more information on the convert command, see ?convert in the help system.

Maple provides several ways to define functions. One way is to use arrow notation (which closely resembles standard mathematical notation for a mapping). You can also use the unapply command, which turns an expression into a function.Define the function JSQuLi5H.f := x -> x^2+1/2 ;Evaluate the function at numeric and symbolic values.f(2);f(a+b);Use the unapply command to turn an expression into a function.g := unapply( x^2 + 1/2, x );Encuentre NiMtSSJnRzYiNiMsJiomSSNQaUdJKnByb3RlY3RlZEdGKiIiIiIiIyEiIkYtSSJhR0YlRi0=Sustituya a por 1 en NiMtSSJnRzYiNiMsJiomSSNQaUdJKnByb3RlY3RlZEdGKiIiIiIiIyEiIkYtSSJhR0YlRi0=. (use subs(a=1, g(-Pi/2-a)) )simplifique el resultado anteriorDefina la funci\363n h(x,y) = NiMqJi1JJHNpbkc2IjYjLCZJInhHRiYiIiJJInlHRiZGKkYqLCYqJEYpIiIjRipGKyEiIkYv y encuentre NiMtSSJoRzYiNiQiIiEqJkkjUGlHSSpwcm90ZWN0ZWRHRioiIiIiIiMhIiI= For more information on the unapply command, see ?-> and ?unapply in the help system.

You can use Maple to solve and verify solutions to equations and systems of equations.

Use Maple to solve the following equation: NiMvLCgqJCklInhHIiIkIiIiRikqKCUiYUdGKSokKUYnIiIjRilGKUYuISIiRi8qKCIjOEYpRixGKUYoRi9GKSwoKipGMUYpRitGKUYnRikiIidGL0YpKigiIzVGKUYnRilGKEYvRikqKCIiJkYpRitGKUYoRi9GLw== .eqn := x^3-1/2*a*x^2+13/3*x^2 = 13/6*a*x+10/3*x-5/3*a;solve( eqn, {x} );To verify one of the solutions, evaluate the equation at that particular value of x.eval( eqn , x=a/2 );Asigne el nombre ecuac a la ecuaci\363n NiMvKiYsJkkieEc2IiIiIiIiJCEiIkYoLCZGJkYoIiImRihGKiwmKiZGKUYoKiRGJiIiI0YoRihGKEYq y luego resuelva For more information on the solve and eval commands, see ?solve and ?eval in the help system.

You can also solve systems of equations in Maple. Consider the following set of four equations and five unknowns:eqn1 := a+2*b+3*c+4*d+5*e=41;eqn2 := 5*a+5*b+4*c+3*d+2*e=20;eqn3 := 3*b+4*c-8*d+2*e=125;eqn4 := a+b+c+d+e=9;Now solve the system for the variables NiMlImFH, NiMlImJH, NiMlImNH, and d. Maple returns its solutions in terms of the fifth variable, e. Since there are four equations and five unknowns, you have a parametric set of solutions. On the other hand, if you solve for a, b, c, d, and e, Maple selects one of the variables (at random) as a free parameter.solve( {eqn1, eqn2, eqn3, eqn4}, {a, b, c, d} );To verify that this solution satisfies NiMlJWVxbjFH and NiMlJWVxbjJH, evaluate both of them at this solution. eval( {eqn1, eqn2} , % );Asigne los nombres ec1 ,ec2 ,ec3 y ec4 a las ecuaciones a+2*b+3*c+4*d=41 , 5*a+5*b+4*c+3*d=20 , 3*b+4*c-8*d=125 y a+b+c+d=9 , respectivamente y luego resuelva el sistema de ecuaciones dado por ellas:

The following examples show other types of equations that you can solve using Maple, such as equations using trigonometric functions and absolute values.Solve an equation involving trigonometric functions.Resolver la ecuaci\363n NiMvLCYtSSdhcmNjb3NHNiRJKnByb3RlY3RlZEdGKEkoX3N5c2xpYkc2IjYjSSJ4R0YqIiIiLUknYXJjdGFuR0YnRishIiIiIiE= :As a final example, solve an equation containing absolute values: NiMvKiQtJSRhYnNHNiMsJiokLCYlInpHIiIiLUYmNiMsJkYrRiwiIiNGLEYsRjBGLEYsISIiRjAiIio=.abs( (z+abs(z+2))^2-1 )^2 = 9;

The following examples show how easy it is to solve systems of inequalities in Maple.Use Maple to solve systems of inequalities, for example, NiMyKiQlInhHIiIjIiIi, NiMxKiQlInlHIiIjIiIi, and NiMyLCYlInhHIiIiJSJ5R0YmKiZGJkYmIiIjISIi. solve( {x^2<1, y^2<=1, x+y<1/2}, {x,y} );This example solves the inequality NiMyLCglInhHIiIiJSJ5R0YmKiYiIiVGJiwmRiVGJkYnRiYhIiJGJiIjNQ== for x in terms of y.ineq := x+y+4/(x+y) < 10:solve( ineq, {x} );Resolver la inecuaci\363n x^2+x>5 :Prueba la siguiente forma solve(x^2+x>5,x) .\277Cual es la diferencia? Not only does Maple solve inequalities, but it can also work with complex inequalities, for example, NiMwKigiIiMiIiItJSVzcXJ0RzYjLCZGJiEiIiUiSUdGK0YmLUYoNiMsJkYmRitGLEYmRiYiIiE=, and compute its Boolean value, by using the is command.expr := 2*sqrt(-1-I)*sqrt(-1+I);is( expr <> 0 );For more information on the is command, see ?is in the help system.