1.) Use Maple to write x5 -6x3 -5x-3x4+18x2+15 as a product of exact linear factors. By exact, I mean you should leave any non-rational factors expressed as radicals; do not approximate terms like p3 as 1.73205, etc. First let us call our polynomial p(x). p:= x -> x^5 -6*x^3 -5*x -3*x^4 + 18*x^2 + 15;

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Now let us find the complete factorization of p(x) with respect to x. with(PolynomialTools): pTemp:= Split(p(x),x);

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However, pTemp is expressed as linear factors with some factors in the form x + RootOf(Polynomial Expression), which is not how we want our output. We thus convert these into radical representation to have our final result: convert(pTemp,radical);

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This is the exact linear factorization of our original polynomial p(x). 2.) Consider the planar curve defined by x2y3 + y2 + y - 2ex = 0. Using only Maple, find the slope of the tangent line to the curve at (0, 1). Then plot the curve and the tangent line on the same graph. Hint: you might want to use implicitplot from the library plots. You might find implicitdiff helpful, too. f:= (x,y) -> x^2 * y^3 + y^2 + y -2*exp(x);

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Find the derivative dy/dx through implicit differentiation, store it as yDerivative yDerivative:= implicitdiff(f(x,y)=0,y,x);

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Now let us find the slope of the tangent line (aka the derivative) at the point of interest (0,1): tangentSlope = eval(yDerivative, [x=0, y=1]);

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Given the tangent slope, let us now state the formula for the tangent line. We can easily do this using point slope form, given that the line passes through the tangency point (0,1). This lets us use the form: (y-y0) = m(x-x0). pointSlopeForm:= y-1 = (2/3) * (x-0); tangentLine:= solve(pointSlopeForm,y);

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with(plots, implicitplot): implicitplot([f(x,y)=0, y = tangentLine], x=-10..10,y=-10..10); implicitplot([f(x,y)=0, y = tangentLine], x=-2..2,y=-2..2);

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KysrK0UiISM7JCIkJT0hIiM3JCQiJEsiISIjJCIyLCsrKysrKyk9ISM7NyQkIiRPIiEiIyQiMm5tbW1tbW0hPiEjOzckJCIlOzkhIiQkIiVXPiEiJDckJCIyLSsrKysrK10iISM7JCIjPyEiIi0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiQiIiEhIiIkIiIhISIiLSYlJl9BWElTRzYjIiIiNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIi0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJiUmX0FYSVNHNiMiIiM2Ji0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIiLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0lK0FYRVNMQUJFTFNHNictSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USJ4NiIvJSdmYW1pbHlHUShERUZBVUxUNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEieTYiLyUnZmFtaWx5R1EoREVGQVVMVDYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii0lJUZPTlRHNiUlKERFRkFVTFRHJShERUZBVUxURyIjNSUrSE9SSVpPTlRBTEclK0hPUklaT05UQUxHLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkUyIhIiItJSlCT1VORFNfWUc2IyQiJD8iISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlIXkkISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJV1QISIiLSUpQ0hJTERSRU5HNiI=

The graph of the planar curve along with the tangent line shows that our line is indeed tangent to the curve at the desired point (0,1). 3.) Plot the function f(x) = 2 sin x - x3 - 5, for x in [-4, 4]. Find all the zeros of the function with an accuracy of 20 decimal digits. Hint: See Digits, fsolve, and/or evalf. Make sure you find all the zeroes. Justify your answer. f(x):= x -> 2 * sin(x) - x^3 - 5;

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First, let us plot our function in the specified range of -4 to 4 to understand and visualize its behavior. By plotting the function over the given range, we can see how many zeroes we are looking for. plot(f(x),-4..4);

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

Looking over our plot, we see that there is only one zero, which should be roughly around x=-2. Now we can confidently use the solve function to derive our single solution for the range -4 to 4, taking care to solve to 20 digits as specified in the problem statement. Digits:= 20; singleSolution := fsolve(f(x),-4..4);

IiM/

JCE1WStSdiZ5d0FIIT4hIz4=

In accordance with our plot and visual estimate, we note that this equation has a single solution in the range -4 to 4. We are confident in our solution and its uniqueness for this range because our graph showed only one solution.