Book a Demo!
CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutPoliciesSign UpSign In
Download

poly zeros

51 views
Tweet
Share
Share

Use this code for the half of the packet called Analyzing and Solving Polynomial Equations. All you need to do is edit line 1 with each new function on the left side of the equation. When you are done with all exercises, hide all the code and upload to Moodle.

f(x)=(x^3-2*x^2-3*x+6)
show('the function is ',f(x))
zeros=solve(f(x)==0,x)
zeros=[zeros[i].rhs() for i in [0..len(zeros)-1]]
show('the zeros are ',zeros)
p=plot(f(x),xmin=min(zeros)-1,xmax=max(zeros)+1)
intercepts=[[i,0]  for i in zeros if imag_part(i)==0]
p+=list_plot(intercepts,size=30,color='red')
p
the function is x3−2 x2−3 x+6\displaystyle x^{3} - 2 \, x^{2} - 3 \, x + 6
the zeros are [−3\displaystyle -\sqrt{3}, 3\displaystyle \sqrt{3}, 2\displaystyle 2]
f(x)=(x^6-2*x^4-4*x^2+8)
show('the function is ',f(x))
zeros=solve(f(x)==0,x)
zeros=[zeros[i].rhs() for i in [0..len(zeros)-1]]
show('the zeros are ',zeros)
intercepts=[[i,0]  for i in zeros if imag_part(i)==0]
p=plot(f(x),xmin=min(zeros)-1,xmax=max(zeros)+1)
intercepts=[[i,0]  for i in zeros if imag_part(i)==0]
p+=list_plot(intercepts,size=30,color='red')
p
the function is x6−2 x4−4 x2+8\displaystyle x^{6} - 2 \, x^{4} - 4 \, x^{2} + 8
the zeros are [−2\displaystyle -\sqrt{2}, 2\displaystyle \sqrt{2}, −i 2\displaystyle -i \, \sqrt{2}, i 2\displaystyle i \, \sqrt{2}]

f(x)=(x^6+5*x^4-4*x^2-20)
show('the function is ',f(x))
zeros=solve(f(x)==0,x)
zeros=[zeros[i].rhs() for i in [0..len(zeros)-1]]
show('the zeros are ',zeros)
p=plot(f(x),xmin=min(zeros)-1,xmax=max(zeros)+1)
intercepts=[[i,0]  for i in zeros if imag_part(i)==0]
p+=list_plot(intercepts,size=30,color='red')
p
the function is x6+5 x4−4 x2−20\displaystyle x^{6} + 5 \, x^{4} - 4 \, x^{2} - 20
the zeros are [−i 5\displaystyle -i \, \sqrt{5}, i 5\displaystyle i \, \sqrt{5}, −i 2\displaystyle -i \, \sqrt{2}, i 2\displaystyle i \, \sqrt{2}, −2\displaystyle -\sqrt{2}, 2\displaystyle \sqrt{2}]
Error in lines 6-6 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags) in namespace, locals File "", line 1, in <module> File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 567, in wrapper return func(*args, **options) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/plot/plot.py", line 1941, in plot G = funcs.plot(*args, **original_opts) File "sage/symbolic/expression.pyx", line 12032, in sage.symbolic.expression.Expression.plot (build/cythonized/sage/symbolic/expression.cpp:68738) return plot(f, *args, **kwds) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 567, in wrapper return func(*args, **options) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/plot/plot.py", line 1956, in plot G = _plot(funcs, (xmin, xmax), **kwds) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/plot/plot.py", line 2077, in _plot funcs, ranges = setup_for_eval_on_grid(funcs, [xrange], options['plot_points']) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/plot/misc.py", line 121, in setup_for_eval_on_grid ranges = [[float(z) for z in r] for r in ranges] File "sage/symbolic/expression.pyx", line 1428, in sage.symbolic.expression.Expression.__float__ (build/cythonized/sage/symbolic/expression.cpp:11836) raise TypeError("unable to simplify to float approximation") TypeError: unable to simplify to float approximation 

f(x)=(x^6+5*x^4-4*x^2-20)
show('the function is ',f(x))
zeros=solve(f(x)==0,x)
zeros=[zeros[i].rhs() for i in [0..len(zeros)-1]]
show('the zeros are ',zeros)
p=plot(f(x),xmin=min(zeros)-1,xmax=max(zeros)+1)
intercepts=[[i,0]  for i in zeros if imag_part(i)==0]
p+=list_plot(intercepts,size=30,color='red')
p
the function is x6+5 x4−4 x2−20\displaystyle x^{6} + 5 \, x^{4} - 4 \, x^{2} - 20
the zeros are [−i 5\displaystyle -i \, \sqrt{5}, i 5\displaystyle i \, \sqrt{5}, −i 2\displaystyle -i \, \sqrt{2}, i 2\displaystyle i \, \sqrt{2}, −2\displaystyle -\sqrt{2}, 2\displaystyle \sqrt{2}]
Error in lines 6-6 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags) in namespace, locals File "", line 1, in <module> File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 567, in wrapper return func(*args, **options) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/plot/plot.py", line 1941, in plot G = funcs.plot(*args, **original_opts) File "sage/symbolic/expression.pyx", line 12032, in sage.symbolic.expression.Expression.plot (build/cythonized/sage/symbolic/expression.cpp:68738) return plot(f, *args, **kwds) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 567, in wrapper return func(*args, **options) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/plot/plot.py", line 1956, in plot G = _plot(funcs, (xmin, xmax), **kwds) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/plot/plot.py", line 2077, in _plot funcs, ranges = setup_for_eval_on_grid(funcs, [xrange], options['plot_points']) File "/ext/sage/sage-8.3_1804/local/lib/python2.7/site-packages/sage/plot/misc.py", line 121, in setup_for_eval_on_grid ranges = [[float(z) for z in r] for r in ranges] File "sage/symbolic/expression.pyx", line 1428, in sage.symbolic.expression.Expression.__float__ (build/cythonized/sage/symbolic/expression.cpp:11836) raise TypeError("unable to simplify to float approximation") TypeError: unable to simplify to float approximation
the function is x8−26 x4+25\displaystyle x^{8} - 26 \, x^{4} + 25
the zeros are [i\displaystyle i, −1\displaystyle -1, −i\displaystyle -i, 1\displaystyle 1, i 5\displaystyle i \, \sqrt{5}, −5\displaystyle -\sqrt{5}, −i 5\displaystyle -i \, \sqrt{5}, 5\displaystyle \sqrt{5}]
the function is x4−5 x2−36\displaystyle x^{4} - 5 \, x^{2} - 36
the zeros are [−2i\displaystyle -2 i, 2i\displaystyle 2 i, −3\displaystyle -3, 3\displaystyle 3]
the function is x3+3 x2−14 x−20\displaystyle x^{3} + 3 \, x^{2} - 14 \, x - 20
the zeros are [−5+1\displaystyle -\sqrt{5} + 1, 5+1\displaystyle \sqrt{5} + 1, −5\displaystyle -5]
the function is x3−2 x2+3 x−6\displaystyle x^{3} - 2 \, x^{2} + 3 \, x - 6
the zeros are [−i 3\displaystyle -i \, \sqrt{3}, i 3\displaystyle i \, \sqrt{3}, 2\displaystyle 2]
the function is x4−14 x2+45\displaystyle x^{4} - 14 \, x^{2} + 45
the zeros are [−5\displaystyle -\sqrt{5}, 5\displaystyle \sqrt{5}, −3\displaystyle -3, 3\displaystyle 3]
the function is x4−6 x2+8\displaystyle x^{4} - 6 \, x^{2} + 8
the zeros are [−2\displaystyle -\sqrt{2}, 2\displaystyle \sqrt{2}, −2\displaystyle -2, 2\displaystyle 2]
the function is x4−3 x2−18\displaystyle x^{4} - 3 \, x^{2} - 18
the zeros are [−i 3\displaystyle -i \, \sqrt{3}, i 3\displaystyle i \, \sqrt{3}, −6\displaystyle -\sqrt{6}, 6\displaystyle \sqrt{6}]
the function is x3−1\displaystyle x^{3} - 1
the zeros are [12i 3−12\displaystyle \frac{1}{2} i \, \sqrt{3} - \frac{1}{2}, −12i 3−12\displaystyle -\frac{1}{2} i \, \sqrt{3} - \frac{1}{2}, 1\displaystyle 1]
the function is x3+3 x2−x−3\displaystyle x^{3} + 3 \, x^{2} - x - 3
the zeros are [1\displaystyle 1, −1\displaystyle -1, −3\displaystyle -3]
the function is x5+2 x4+11 x3+22 x2+24 x+48\displaystyle x^{5} + 2 \, x^{4} + 11 \, x^{3} + 22 \, x^{2} + 24 \, x + 48
the zeros are [−2i 2\displaystyle -2 i \, \sqrt{2}, 2i 2\displaystyle 2 i \, \sqrt{2}, −2\displaystyle -2, −i 3\displaystyle -i \, \sqrt{3}, i 3\displaystyle i \, \sqrt{3}]
the function is x3−2 x2−3 x+6\displaystyle x^{3} - 2 \, x^{2} - 3 \, x + 6
the zeros are [−3\displaystyle -\sqrt{3}, 3\displaystyle \sqrt{3}, 2\displaystyle 2]
the function is x6−2 x4−4 x2+8\displaystyle x^{6} - 2 \, x^{4} - 4 \, x^{2} + 8
the zeros are [−2\displaystyle -\sqrt{2}, 2\displaystyle \sqrt{2}, −i 2\displaystyle -i \, \sqrt{2}, i 2\displaystyle i \, \sqrt{2}]