Find approximations to within 10^-5 to all the zeros of each of the following po

Question

Find approximations to within 10^-5 to all the zeros of each of the following polynomials by first finding the real zeros using Newton's method and then reducing to polynomials of lower degree to determine any complex zeros. f (x) = x^4 + 5x^3 - 9x^2 - 85x - 136 f (x) = x^4 - 2x^3 - 12 x^2 + 16x - 40 f (x) = x^4 + x^3 + 3x^2 + 2x + 2 f (x) = x^5 + 11x^4 - 21x^3 - 10x^2 - 21x - 5 f (x) = 16x^4 + 88x^3 + 159x^2 + 76x - 240 f (x) = x^4 - 4x^2 - 3x + 5 f (x) x^4 - 2x^3 - 4x^2 + 4x + 4 f (x) = x^3 - 7x^2 + 14x - 6 Please show a Maple code for this problem.

Explanation / Answer

F:= x-> 2*(x^2) – 4*cos(5*x) – 4*x*sin(x) + 1;

Fd:= diff(F(x),x);

G:= unapply(x - F(x)/Fd, x);

x[0]:= 0;

x[1]:= evalf(G(x[0]));

for i from 2 to 6 do

x[i]:= evalf(G(x[i-1]));

end do;

for i from 1 to 6 do

if abs(x[i] – x[i-1]) > 0.00005
print("accuracy not reached")
else print ("accuracy reached")
end if;
end do;

or direct inbuilt function

NewtonsMethod(f(x), x = a, opts)

example a)

NewtonsMethod(x^4+5x^3-9x^2-85x-136, x = 0, output=5)

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