What is the general solution for cos^{23}(x)+cos^2(x)=0 ?

The general solution for cos^{23}(x)+cos^2(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin,x=pi+2pin

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cos^{23}(x)+cos^2(x)=0

Solution

cos^{23} (x) + cos^{2} (x) = 0

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = π + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

cos^{23} (x) + cos^{2} (x) = 0

Solve by substitution

cos^{23} (x) + cos^{2} (x) = 0

Let:

cos (x) = u

u^{23} + u^{2} = 0

u^{23} + u^{2} = 0 : u = 0, u = - 1

u^{23} + u^{2} = 0

Factor

u^{23} + u^{2} : u^{2} (u + 1) (u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1) (u^{14} - u^{7} + 1)

u^{23} + u^{2}

Factor out common term

u^{2} : u^{2} (u^{21} + 1)

u^{23} + u^{2}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{23} = u^{21} u^{2}

= u^{21} u^{2} + u^{2}

Factor out common term

u^{2}

= u^{2} (u^{21} + 1)

= u^{2} (u^{21} + 1)

Factor

u^{21} + 1 : (u + 1) (u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1) (u^{14} - u^{7} + 1)

u^{21} + 1

Rewrite

u^{21} + 1

(u^{7})^{3} + 1^{3}

u^{21} + 1

Rewrite

1

1^{3}

= u^{21} + 1^{3}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

u^{21} = (u^{7})^{3}

= (u^{7})^{3} + 1^{3}

= (u^{7})^{3} + 1^{3}

Apply Sum of Cubes Formula:

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

(u^{7})^{3} + 1^{3} = (u^{7} + 1) (u^{14} - u^{7} + 1)

= (u^{7} + 1) (u^{14} - u^{7} + 1)

Factor

u^{7} + 1 : (u + 1) (u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1)

u^{7} + 1

Rewrite

1

1^{7}

= u^{7} + 1^{7}

Apply factoring rule:

x^{n} + y^{n} = (x + y) (x^{n - 1} - x^{n - 2} y + \dots - x y^{n - 2} + y^{n - 1})

n is odd

u^{7} + 1^{7} = (u + 1) (u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1)

= (u + 1) (u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1)

= (u + 1) (u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1) (u^{14} - u^{7} + 1)

= u^{2} (u + 1) (u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1) (u^{14} - u^{7} + 1)

u^{2} (u + 1) (u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1) (u^{14} - u^{7} + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or u + 1 = 0 or u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1 = 0 or u^{14} - u^{7} + 1 = 0

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

Solve

u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1 = 0 :

No Solution for

u \in R

u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1 = 0

Find one solution for

u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1 = 0

using Newton-Raphson:

No Solution for

u \in R

u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1 = 0

Newton-Raphson Approximation Definition

f (u) = u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1

Find

f^{^{'}} (u) : 6 u^{5} - 5 u^{4} + 4 u^{3} - 3 u^{2} + 2 u - 1

\frac{d}{d u} (u^{6} - u^{5} + u^{4} - u^{3} + u^{2} - u + 1)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{6}) - \frac{d}{d u} (u^{5}) + \frac{d}{d u} (u^{4}) - \frac{d}{d u} (u^{3}) + \frac{d}{d u} (u^{2}) - \frac{d u}{d u} + \frac{d}{d u} (1)

\frac{d}{d u} (u^{6}) = 6 u^{5}

\frac{d}{d u} (u^{6})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 6 u^{6 - 1}

Simplify

= 6 u^{5}

\frac{d}{d u} (u^{5}) = 5 u^{4}

\frac{d}{d u} (u^{5})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 5 u^{5 - 1}

Simplify

= 5 u^{4}

\frac{d}{d u} (u^{4}) = 4 u^{3}

\frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 u^{4 - 1}

Simplify

= 4 u^{3}

\frac{d}{d u} (u^{3}) = 3 u^{2}

\frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 u^{3 - 1}

Simplify

= 3 u^{2}

\frac{d}{d u} (u^{2}) = 2 u

\frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 u^{2 - 1}

Simplify

= 2 u

\frac{d u}{d u} = 1

\frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 1

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 6 u^{5} - 5 u^{4} + 4 u^{3} - 3 u^{2} + 2 u - 1 + 0

Simplify

= 6 u^{5} - 5 u^{4} + 4 u^{3} - 3 u^{2} + 2 u - 1

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.66666 \dots : Δ u_{1} = 0.33333 \dots

f (u_{0}) = 1^{6} - 1^{5} + 1^{4} - 1^{3} + 1^{2} - 1 + 1 = 1

f^{^{'}} (u_{0}) = 6 \cdot 1^{5} - 5 \cdot 1^{4} + 4 \cdot 1^{3} - 3 \cdot 1^{2} + 2 \cdot 1 - 1 = 3

u_{1} = 0.66666 \dots

Δ u_{1} = ∣ 0.66666 \dots - 1 ∣ = 0.33333 \dots

Δ u_{1} = 0.33333 \dots

u_{2} = 52.11111 \dots : Δ u_{2} = 51.44444 \dots

f (u_{1}) = 0.66666 \dots^{6} - 0.66666 \dots^{5} + 0.66666 \dots^{4} - 0.66666 \dots^{3} + 0.66666 \dots^{2} - 0.66666 \dots + 1 = 0.63511 \dots

f^{^{'}} (u_{1}) = 6 \cdot 0.66666 \dots^{5} - 5 \cdot 0.66666 \dots^{4} + 4 \cdot 0.66666 \dots^{3} - 3 \cdot 0.66666 \dots^{2} + 2 \cdot 0.66666 \dots - 1 = - 0.01234 \dots

u_{2} = 52.11111 \dots

Δ u_{2} = ∣ 52.11111 \dots - 0.66666 \dots ∣ = 51.44444 \dots

Δ u_{2} = 51.44444 \dots

u_{3} = 43.45309 \dots : Δ u_{3} = 8.65801 \dots

f (u_{2}) = 52.11111 \dots^{6} - 52.11111 \dots^{5} + 52.11111 \dots^{4} - 52.11111 \dots^{3} + 52.11111 \dots^{2} - 52.11111 \dots + 1 = 19648388910.5653

f^{^{'}} (u_{2}) = 6 \cdot 52.11111 \dots^{5} - 5 \cdot 52.11111 \dots^{4} + 4 \cdot 52.11111 \dots^{3} - 3 \cdot 52.11111 \dots^{2} + 2 \cdot 52.11111 \dots - 1 = 2269387078.62673 \dots

u_{3} = 43.45309 \dots

Δ u_{3} = ∣ 43.45309 \dots - 52.11111 \dots ∣ = 8.65801 \dots

Δ u_{3} = 8.65801 \dots

u_{4} = 36.23796 \dots : Δ u_{4} = 7.21513 \dots

f (u_{3}) = 43.45309 \dots^{6} - 43.45309 \dots^{5} + 43.45309 \dots^{4} - 43.45309 \dots^{3} + 43.45309 \dots^{2} - 43.45309 \dots + 1 = 6580259602.39668 \dots

f^{^{'}} (u_{3}) = 6 \cdot 43.45309 \dots^{5} - 5 \cdot 43.45309 \dots^{4} + 4 \cdot 43.45309 \dots^{3} - 3 \cdot 43.45309 \dots^{2} + 2 \cdot 43.45309 \dots - 1 = 912008321.82339 \dots

u_{4} = 36.23796 \dots

Δ u_{4} = ∣ 36.23796 \dots - 43.45309 \dots ∣ = 7.21513 \dots

Δ u_{4} = 7.21513 \dots

u_{5} = 30.22521 \dots : Δ u_{5} = 6.01274 \dots

f (u_{4}) = 36.23796 \dots^{6} - 36.23796 \dots^{5} + 36.23796 \dots^{4} - 36.23796 \dots^{3} + 36.23796 \dots^{2} - 36.23796 \dots + 1 = 2203741351.76969 \dots

f^{^{'}} (u_{4}) = 6 \cdot 36.23796 \dots^{5} - 5 \cdot 36.23796 \dots^{4} + 4 \cdot 36.23796 \dots^{3} - 3 \cdot 36.23796 \dots^{2} + 2 \cdot 36.23796 \dots - 1 = 366511428.47054 \dots

u_{5} = 30.22521 \dots

Δ u_{5} = ∣ 30.22521 \dots - 36.23796 \dots ∣ = 6.01274 \dots

Δ u_{5} = 6.01274 \dots

u_{6} = 25.21442 \dots : Δ u_{6} = 5.01079 \dots

f (u_{5}) = 30.22521 \dots^{6} - 30.22521 \dots^{5} + 30.22521 \dots^{4} - 30.22521 \dots^{3} + 30.22521 \dots^{2} - 30.22521 \dots + 1 = 738040770.05592 \dots

f^{^{'}} (u_{5}) = 6 \cdot 30.22521 \dots^{5} - 5 \cdot 30.22521 \dots^{4} + 4 \cdot 30.22521 \dots^{3} - 3 \cdot 30.22521 \dots^{2} + 2 \cdot 30.22521 \dots - 1 = 147290289.66438 \dots

u_{6} = 25.21442 \dots

Δ u_{6} = ∣ 25.21442 \dots - 30.22521 \dots ∣ = 5.01079 \dots

Δ u_{6} = 5.01079 \dots

u_{7} = 21.03856 \dots : Δ u_{7} = 4.17585 \dots

f (u_{6}) = 25.21442 \dots^{6} - 25.21442 \dots^{5} + 25.21442 \dots^{4} - 25.21442 \dots^{3} + 25.21442 \dots^{2} - 25.21442 \dots + 1 = 247174180.13704 \dots

f^{^{'}} (u_{6}) = 6 \cdot 25.21442 \dots^{5} - 5 \cdot 25.21442 \dots^{4} + 4 \cdot 25.21442 \dots^{3} - 3 \cdot 25.21442 \dots^{2} + 2 \cdot 25.21442 \dots - 1 = 59191278.12486 \dots

u_{7} = 21.03856 \dots

Δ u_{7} = ∣ 21.03856 \dots - 25.21442 \dots ∣ = 4.17585 \dots

Δ u_{7} = 4.17585 \dots

u_{8} = 17.55845 \dots : Δ u_{8} = 3.48010 \dots

f (u_{7}) = 21.03856 \dots^{6} - 21.03856 \dots^{5} + 21.03856 \dots^{4} - 21.03856 \dots^{3} + 21.03856 \dots^{2} - 21.03856 \dots + 1 = 82780889.58008 \dots

f^{^{'}} (u_{7}) = 6 \cdot 21.03856 \dots^{5} - 5 \cdot 21.03856 \dots^{4} + 4 \cdot 21.03856 \dots^{3} - 3 \cdot 21.03856 \dots^{2} + 2 \cdot 21.03856 \dots - 1 = 23786860.21097 \dots

u_{8} = 17.55845 \dots

Δ u_{8} = ∣ 17.55845 \dots - 21.03856 \dots ∣ = 3.48010 \dots

Δ u_{8} = 3.48010 \dots

u_{9} = 14.65809 \dots : Δ u_{9} = 2.90036 \dots

f (u_{8}) = 17.55845 \dots^{6} - 17.55845 \dots^{5} + 17.55845 \dots^{4} - 17.55845 \dots^{3} + 17.55845 \dots^{2} - 17.55845 \dots + 1 = 27724453.98017 \dots

f^{^{'}} (u_{8}) = 6 \cdot 17.55845 \dots^{5} - 5 \cdot 17.55845 \dots^{4} + 4 \cdot 17.55845 \dots^{3} - 3 \cdot 17.55845 \dots^{2} + 2 \cdot 17.55845 \dots - 1 = 9558960.37202 \dots

u_{9} = 14.65809 \dots

Δ u_{9} = ∣ 14.65809 \dots - 17.55845 \dots ∣ = 2.90036 \dots

Δ u_{9} = 2.90036 \dots

u_{10} = 12.24081 \dots : Δ u_{10} = 2.41728 \dots

f (u_{9}) = 14.65809 \dots^{6} - 14.65809 \dots^{5} + 14.65809 \dots^{4} - 14.65809 \dots^{3} + 14.65809 \dots^{2} - 14.65809 \dots + 1 = 9285475.65063 \dots

f^{^{'}} (u_{9}) = 6 \cdot 14.65809 \dots^{5} - 5 \cdot 14.65809 \dots^{4} + 4 \cdot 14.65809 \dots^{3} - 3 \cdot 14.65809 \dots^{2} + 2 \cdot 14.65809 \dots - 1 = 3841280.89299 \dots

u_{10} = 12.24081 \dots

Δ u_{10} = ∣ 12.24081 \dots - 14.65809 \dots ∣ = 2.41728 \dots

Δ u_{10} = 2.41728 \dots

u_{11} = 10.22603 \dots : Δ u_{11} = 2.01477 \dots

f (u_{10}) = 12.24081 \dots^{6} - 12.24081 \dots^{5} + 12.24081 \dots^{4} - 12.24081 \dots^{3} + 12.24081 \dots^{2} - 12.24081 \dots + 1 = 3109973.57380 \dots

f^{^{'}} (u_{10}) = 6 \cdot 12.24081 \dots^{5} - 5 \cdot 12.24081 \dots^{4} + 4 \cdot 12.24081 \dots^{3} - 3 \cdot 12.24081 \dots^{2} + 2 \cdot 12.24081 \dots - 1 = 1543583.94342 \dots

u_{11} = 10.22603 \dots

Δ u_{11} = ∣ 10.22603 \dots - 12.24081 \dots ∣ = 2.01477 \dots

Δ u_{11} = 2.01477 \dots

u_{12} = 8.54662 \dots : Δ u_{12} = 1.67940 \dots

f (u_{11}) = 10.22603 \dots^{6} - 10.22603 \dots^{5} + 10.22603 \dots^{4} - 10.22603 \dots^{3} + 10.22603 \dots^{2} - 10.22603 \dots + 1 = 1041657.31792 \dots

f^{^{'}} (u_{11}) = 6 \cdot 10.22603 \dots^{5} - 5 \cdot 10.22603 \dots^{4} + 4 \cdot 10.22603 \dots^{3} - 3 \cdot 10.22603 \dots^{2} + 2 \cdot 10.22603 \dots - 1 = 620253.30227 \dots

u_{12} = 8.54662 \dots

Δ u_{12} = ∣ 8.54662 \dots - 10.22603 \dots ∣ = 1.67940 \dots

Δ u_{12} = 1.67940 \dots

u_{13} = 7.14663 \dots : Δ u_{13} = 1.39999 \dots

f (u_{12}) = 8.54662 \dots^{6} - 8.54662 \dots^{5} + 8.54662 \dots^{4} - 8.54662 \dots^{3} + 8.54662 \dots^{2} - 8.54662 \dots + 1 = 348910.71727 \dots

f^{^{'}} (u_{12}) = 6 \cdot 8.54662 \dots^{5} - 5 \cdot 8.54662 \dots^{4} + 4 \cdot 8.54662 \dots^{3} - 3 \cdot 8.54662 \dots^{2} + 2 \cdot 8.54662 \dots - 1 = 249222.40253 \dots

u_{13} = 7.14663 \dots

Δ u_{13} = ∣ 7.14663 \dots - 8.54662 \dots ∣ = 1.39999 \dots

Δ u_{13} = 1.39999 \dots

u_{14} = 5.97940 \dots : Δ u_{14} = 1.16722 \dots

f (u_{13}) = 7.14663 \dots^{6} - 7.14663 \dots^{5} + 7.14663 \dots^{4} - 7.14663 \dots^{3} + 7.14663 \dots^{2} - 7.14663 \dots + 1 = 116877.91488 \dots

f^{^{'}} (u_{13}) = 6 \cdot 7.14663 \dots^{5} - 5 \cdot 7.14663 \dots^{4} + 4 \cdot 7.14663 \dots^{3} - 3 \cdot 7.14663 \dots^{2} + 2 \cdot 7.14663 \dots - 1 = 100132.95261 \dots

u_{14} = 5.97940 \dots

Δ u_{14} = ∣ 5.97940 \dots - 7.14663 \dots ∣ = 1.16722 \dots

Δ u_{14} = 1.16722 \dots

u_{15} = 5.00607 \dots : Δ u_{15} = 0.97332 \dots

f (u_{14}) = 5.97940 \dots^{6} - 5.97940 \dots^{5} + 5.97940 \dots^{4} - 5.97940 \dots^{3} + 5.97940 \dots^{2} - 5.97940 \dots + 1 = 39155.16368 \dots

f^{^{'}} (u_{14}) = 6 \cdot 5.97940 \dots^{5} - 5 \cdot 5.97940 \dots^{4} + 4 \cdot 5.97940 \dots^{3} - 3 \cdot 5.97940 \dots^{2} + 2 \cdot 5.97940 \dots - 1 = 40228.09525 \dots

u_{15} = 5.00607 \dots

Δ u_{15} = ∣ 5.00607 \dots - 5.97940 \dots ∣ = 0.97332 \dots

Δ u_{15} = 0.97332 \dots

u_{16} = 4.19424 \dots : Δ u_{16} = 0.81183 \dots

f (u_{15}) = 5.00607 \dots^{6} - 5.00607 \dots^{5} + 5.00607 \dots^{4} - 5.00607 \dots^{3} + 5.00607 \dots^{2} - 5.00607 \dots + 1 = 13118.88548 \dots

f^{^{'}} (u_{15}) = 6 \cdot 5.00607 \dots^{5} - 5 \cdot 5.00607 \dots^{4} + 4 \cdot 5.00607 \dots^{3} - 3 \cdot 5.00607 \dots^{2} + 2 \cdot 5.00607 \dots - 1 = 16159.64494 \dots

u_{16} = 4.19424 \dots

Δ u_{16} = ∣ 4.19424 \dots - 5.00607 \dots ∣ = 0.81183 \dots

Δ u_{16} = 0.81183 \dots

u_{17} = 3.51690 \dots : Δ u_{17} = 0.67734 \dots

f (u_{16}) = 4.19424 \dots^{6} - 4.19424 \dots^{5} + 4.19424 \dots^{4} - 4.19424 \dots^{3} + 4.19424 \dots^{2} - 4.19424 \dots + 1 = 4396.16496 \dots

f^{^{'}} (u_{16}) = 6 \cdot 4.19424 \dots^{5} - 5 \cdot 4.19424 \dots^{4} + 4 \cdot 4.19424 \dots^{3} - 3 \cdot 4.19424 \dots^{2} + 2 \cdot 4.19424 \dots - 1 = 6490.31866 \dots

u_{17} = 3.51690 \dots

Δ u_{17} = ∣ 3.51690 \dots - 4.19424 \dots ∣ = 0.67734 \dots

Δ u_{17} = 0.67734 \dots

u_{18} = 2.95151 \dots : Δ u_{18} = 0.56538 \dots

f (u_{17}) = 3.51690 \dots^{6} - 3.51690 \dots^{5} + 3.51690 \dots^{4} - 3.51690 \dots^{3} + 3.51690 \dots^{2} - 3.51690 \dots + 1 = 1473.49363 \dots

f^{^{'}} (u_{17}) = 6 \cdot 3.51690 \dots^{5} - 5 \cdot 3.51690 \dots^{4} + 4 \cdot 3.51690 \dots^{3} - 3 \cdot 3.51690 \dots^{2} + 2 \cdot 3.51690 \dots - 1 = 2606.16404 \dots

u_{18} = 2.95151 \dots

Δ u_{18} = ∣ 2.95151 \dots - 3.51690 \dots ∣ = 0.56538 \dots

Δ u_{18} = 0.56538 \dots

u_{19} = 2.47923 \dots : Δ u_{19} = 0.47228 \dots

f (u_{18}) = 2.95151 \dots^{6} - 2.95151 \dots^{5} + 2.95151 \dots^{4} - 2.95151 \dots^{3} + 2.95151 \dots^{2} - 2.95151 \dots + 1 = 494.05485 \dots

f^{^{'}} (u_{18}) = 6 \cdot 2.95151 \dots^{5} - 5 \cdot 2.95151 \dots^{4} + 4 \cdot 2.95151 \dots^{3} - 3 \cdot 2.95151 \dots^{2} + 2 \cdot 2.95151 \dots - 1 = 1046.10186 \dots

u_{19} = 2.47923 \dots

Δ u_{19} = ∣ 2.47923 \dots - 2.95151 \dots ∣ = 0.47228 \dots

Δ u_{19} = 0.47228 \dots

u_{20} = 2.08415 \dots : Δ u_{20} = 0.39507 \dots

f (u_{19}) = 2.47923 \dots^{6} - 2.47923 \dots^{5} + 2.47923 \dots^{4} - 2.47923 \dots^{3} + 2.47923 \dots^{2} - 2.47923 \dots + 1 = 165.76521 \dots

f^{^{'}} (u_{19}) = 6 \cdot 2.47923 \dots^{5} - 5 \cdot 2.47923 \dots^{4} + 4 \cdot 2.47923 \dots^{3} - 3 \cdot 2.47923 \dots^{2} + 2 \cdot 2.47923 \dots - 1 = 419.57444 \dots

u_{20} = 2.08415 \dots

Δ u_{20} = ∣ 2.08415 \dots - 2.47923 \dots ∣ = 0.39507 \dots

Δ u_{20} = 0.39507 \dots

u_{21} = 1.75246 \dots : Δ u_{21} = 0.33168 \dots

f (u_{20}) = 2.08415 \dots^{6} - 2.08415 \dots^{5} + 2.08415 \dots^{4} - 2.08415 \dots^{3} + 2.08415 \dots^{2} - 2.08415 \dots + 1 = 55.70695 \dots

f^{^{'}} (u_{20}) = 6 \cdot 2.08415 \dots^{5} - 5 \cdot 2.08415 \dots^{4} + 4 \cdot 2.08415 \dots^{3} - 3 \cdot 2.08415 \dots^{2} + 2 \cdot 2.08415 \dots - 1 = 167.95023 \dots

u_{21} = 1.75246 \dots

Δ u_{21} = ∣ 1.75246 \dots - 2.08415 \dots ∣ = 0.33168 \dots

Δ u_{21} = 0.33168 \dots

u_{22} = 1.47108 \dots : Δ u_{22} = 0.28138 \dots

f (u_{21}) = 1.75246 \dots^{6} - 1.75246 \dots^{5} + 1.75246 \dots^{4} - 1.75246 \dots^{3} + 1.75246 \dots^{2} - 1.75246 \dots + 1 = 18.80617 \dots

f^{^{'}} (u_{21}) = 6 \cdot 1.75246 \dots^{5} - 5 \cdot 1.75246 \dots^{4} + 4 \cdot 1.75246 \dots^{3} - 3 \cdot 1.75246 \dots^{2} + 2 \cdot 1.75246 \dots - 1 = 66.83509 \dots

u_{22} = 1.47108 \dots

Δ u_{22} = ∣ 1.47108 \dots - 1.75246 \dots ∣ = 0.28138 \dots

Δ u_{22} = 0.28138 \dots

u_{23} = 1.22445 \dots : Δ u_{23} = 0.24663 \dots

f (u_{22}) = 1.47108 \dots^{6} - 1.47108 \dots^{5} + 1.47108 \dots^{4} - 1.47108 \dots^{3} + 1.47108 \dots^{2} - 1.47108 \dots + 1 = 6.43831 \dots

f^{^{'}} (u_{22}) = 6 \cdot 1.47108 \dots^{5} - 5 \cdot 1.47108 \dots^{4} + 4 \cdot 1.47108 \dots^{3} - 3 \cdot 1.47108 \dots^{2} + 2 \cdot 1.47108 \dots - 1 = 26.10492 \dots

u_{23} = 1.22445 \dots

Δ u_{23} = ∣ 1.22445 \dots - 1.47108 \dots ∣ = 0.24663 \dots

Δ u_{23} = 0.24663 \dots

u_{24} = 0.98361 \dots : Δ u_{24} = 0.24083 \dots

f (u_{23}) = 1.22445 \dots^{6} - 1.22445 \dots^{5} + 1.22445 \dots^{4} - 1.22445 \dots^{3} + 1.22445 \dots^{2} - 1.22445 \dots + 1 = 2.30467 \dots

f^{^{'}} (u_{23}) = 6 \cdot 1.22445 \dots^{5} - 5 \cdot 1.22445 \dots^{4} + 4 \cdot 1.22445 \dots^{3} - 3 \cdot 1.22445 \dots^{2} + 2 \cdot 1.22445 \dots - 1 = 9.56939 \dots

u_{24} = 0.98361 \dots

Δ u_{24} = ∣ 0.98361 \dots - 1.22445 \dots ∣ = 0.24083 \dots

Δ u_{24} = 0.24083 \dots

u_{25} = 0.63257 \dots : Δ u_{25} = 0.35104 \dots

f (u_{24}) = 0.98361 \dots^{6} - 0.98361 \dots^{5} + 0.98361 \dots^{4} - 0.98361 \dots^{3} + 0.98361 \dots^{2} - 0.98361 \dots + 1 = 0.95320 \dots

f^{^{'}} (u_{24}) = 6 \cdot 0.98361 \dots^{5} - 5 \cdot 0.98361 \dots^{4} + 4 \cdot 0.98361 \dots^{3} - 3 \cdot 0.98361 \dots^{2} + 2 \cdot 0.98361 \dots - 1 = 2.71536 \dots

u_{25} = 0.63257 \dots

Δ u_{25} = ∣ 0.63257 \dots - 0.98361 \dots ∣ = 0.35104 \dots

Δ u_{25} = 0.35104 \dots

u_{26} = 6.14224 \dots : Δ u_{26} = 5.50967 \dots

f (u_{25}) = 0.63257 \dots^{6} - 0.63257 \dots^{5} + 0.63257 \dots^{4} - 0.63257 \dots^{3} + 0.63257 \dots^{2} - 0.63257 \dots + 1 = 0.63735 \dots

f^{^{'}} (u_{25}) = 6 \cdot 0.63257 \dots^{5} - 5 \cdot 0.63257 \dots^{4} + 4 \cdot 0.63257 \dots^{3} - 3 \cdot 0.63257 \dots^{2} + 2 \cdot 0.63257 \dots - 1 = - 0.11567 \dots

u_{26} = 6.14224 \dots

Δ u_{26} = ∣ 6.14224 \dots - 0.63257 \dots ∣ = 5.50967 \dots

Δ u_{26} = 5.50967 \dots

u_{27} = 5.14187 \dots : Δ u_{27} = 1.00036 \dots

f (u_{26}) = 6.14224 \dots^{6} - 6.14224 \dots^{5} + 6.14224 \dots^{4} - 6.14224 \dots^{3} + 6.14224 \dots^{2} - 6.14224 \dots + 1 = 46180.38876 \dots

f^{^{'}} (u_{26}) = 6 \cdot 6.14224 \dots^{5} - 5 \cdot 6.14224 \dots^{4} + 4 \cdot 6.14224 \dots^{3} - 3 \cdot 6.14224 \dots^{2} + 2 \cdot 6.14224 \dots - 1 = 46163.42164 \dots

u_{27} = 5.14187 \dots

Δ u_{27} = ∣ 5.14187 \dots - 6.14224 \dots ∣ = 1.00036 \dots

Δ u_{27} = 1.00036 \dots

u_{28} = 4.30753 \dots : Δ u_{28} = 0.83434 \dots

f (u_{27}) = 5.14187 \dots^{6} - 5.14187 \dots^{5} + 5.14187 \dots^{4} - 5.14187 \dots^{3} + 5.14187 \dots^{2} - 5.14187 \dots + 1 = 15472.36679 \dots

f^{^{'}} (u_{27}) = 6 \cdot 5.14187 \dots^{5} - 5 \cdot 5.14187 \dots^{4} + 4 \cdot 5.14187 \dots^{3} - 3 \cdot 5.14187 \dots^{2} + 2 \cdot 5.14187 \dots - 1 = 18544.23303 \dots

u_{28} = 4.30753 \dots

Δ u_{28} = ∣ 4.30753 \dots - 5.14187 \dots ∣ = 0.83434 \dots

Δ u_{28} = 0.83434 \dots

u_{29} = 3.61143 \dots : Δ u_{29} = 0.69609 \dots

f (u_{28}) = 4.30753 \dots^{6} - 4.30753 \dots^{5} + 4.30753 \dots^{4} - 4.30753 \dots^{3} + 4.30753 \dots^{2} - 4.30753 \dots + 1 = 5184.67948 \dots

f^{^{'}} (u_{28}) = 6 \cdot 4.30753 \dots^{5} - 5 \cdot 4.30753 \dots^{4} + 4 \cdot 4.30753 \dots^{3} - 3 \cdot 4.30753 \dots^{2} + 2 \cdot 4.30753 \dots - 1 = 7448.26072 \dots

u_{29} = 3.61143 \dots

Δ u_{29} = ∣ 3.61143 \dots - 4.30753 \dots ∣ = 0.69609 \dots

Δ u_{29} = 0.69609 \dots

u_{30} = 3.03044 \dots : Δ u_{30} = 0.58099 \dots

f (u_{29}) = 3.61143 \dots^{6} - 3.61143 \dots^{5} + 3.61143 \dots^{4} - 3.61143 \dots^{3} + 3.61143 \dots^{2} - 3.61143 \dots + 1 = 1737.71673 \dots

f^{^{'}} (u_{29}) = 6 \cdot 3.61143 \dots^{5} - 5 \cdot 3.61143 \dots^{4} + 4 \cdot 3.61143 \dots^{3} - 3 \cdot 3.61143 \dots^{2} + 2 \cdot 3.61143 \dots - 1 = 2990.94430 \dots

u_{30} = 3.03044 \dots

Δ u_{30} = ∣ 3.03044 \dots - 3.61143 \dots ∣ = 0.58099 \dots

Δ u_{30} = 0.58099 \dots

u_{31} = 2.54519 \dots : Δ u_{31} = 0.48524 \dots

f (u_{30}) = 3.03044 \dots^{6} - 3.03044 \dots^{5} + 3.03044 \dots^{4} - 3.03044 \dots^{3} + 3.03044 \dots^{2} - 3.03044 \dots + 1 = 582.60893 \dots

f^{^{'}} (u_{30}) = 6 \cdot 3.03044 \dots^{5} - 5 \cdot 3.03044 \dots^{4} + 4 \cdot 3.03044 \dots^{3} - 3 \cdot 3.03044 \dots^{2} + 2 \cdot 3.03044 \dots - 1 = 1200.63833 \dots

u_{31} = 2.54519 \dots

Δ u_{31} = ∣ 2.54519 \dots - 3.03044 \dots ∣ = 0.48524 \dots

Δ u_{31} = 0.48524 \dots

u_{32} = 2.13939 \dots : Δ u_{32} = 0.40580 \dots

f (u_{31}) = 2.54519 \dots^{6} - 2.54519 \dots^{5} + 2.54519 \dots^{4} - 2.54519 \dots^{3} + 2.54519 \dots^{2} - 2.54519 \dots + 1 = 195.44997 \dots

f^{^{'}} (u_{31}) = 6 \cdot 2.54519 \dots^{5} - 5 \cdot 2.54519 \dots^{4} + 4 \cdot 2.54519 \dots^{3} - 3 \cdot 2.54519 \dots^{2} + 2 \cdot 2.54519 \dots - 1 = 481.63531 \dots

u_{32} = 2.13939 \dots

Δ u_{32} = ∣ 2.13939 \dots - 2.54519 \dots ∣ = 0.40580 \dots

Δ u_{32} = 0.40580 \dots

u_{33} = 1.79897 \dots : Δ u_{33} = 0.34041 \dots

f (u_{32}) = 2.13939 \dots^{6} - 2.13939 \dots^{5} + 2.13939 \dots^{4} - 2.13939 \dots^{3} + 2.13939 \dots^{2} - 2.13939 \dots + 1 = 65.65954 \dots

f^{^{'}} (u_{32}) = 6 \cdot 2.13939 \dots^{5} - 5 \cdot 2.13939 \dots^{4} + 4 \cdot 2.13939 \dots^{3} - 3 \cdot 2.13939 \dots^{2} + 2 \cdot 2.13939 \dots - 1 = 192.87838 \dots

u_{33} = 1.79897 \dots

Δ u_{33} = ∣ 1.79897 \dots - 2.13939 \dots ∣ = 0.34041 \dots

Δ u_{33} = 0.34041 \dots

u_{34} = 1.51087 \dots : Δ u_{34} = 0.28809 \dots

f (u_{33}) = 1.79897 \dots^{6} - 1.79897 \dots^{5} + 1.79897 \dots^{4} - 1.79897 \dots^{3} + 1.79897 \dots^{2} - 1.79897 \dots + 1 = 22.14298 \dots

f^{^{'}} (u_{33}) = 6 \cdot 1.79897 \dots^{5} - 5 \cdot 1.79897 \dots^{4} + 4 \cdot 1.79897 \dots^{3} - 3 \cdot 1.79897 \dots^{2} + 2 \cdot 1.79897 \dots - 1 = 76.85953 \dots

u_{34} = 1.51087 \dots

Δ u_{34} = ∣ 1.51087 \dots - 1.79897 \dots ∣ = 0.28809 \dots

Δ u_{34} = 0.28809 \dots

u_{35} = 1.26028 \dots : Δ u_{35} = 0.25058 \dots

f (u_{34}) = 1.51087 \dots^{6} - 1.51087 \dots^{5} + 1.51087 \dots^{4} - 1.51087 \dots^{3} + 1.51087 \dots^{2} - 1.51087 \dots + 1 = 7.55598 \dots

f^{^{'}} (u_{34}) = 6 \cdot 1.51087 \dots^{5} - 5 \cdot 1.51087 \dots^{4} + 4 \cdot 1.51087 \dots^{3} - 3 \cdot 1.51087 \dots^{2} + 2 \cdot 1.51087 \dots - 1 = 30.15294 \dots

u_{35} = 1.26028 \dots

Δ u_{35} = ∣ 1.26028 \dots - 1.51087 \dots ∣ = 0.25058 \dots

Δ u_{35} = 0.25058 \dots

u_{36} = 1.02183 \dots : Δ u_{36} = 0.23844 \dots

f (u_{35}) = 1.26028 \dots^{6} - 1.26028 \dots^{5} + 1.26028 \dots^{4} - 1.26028 \dots^{3} + 1.26028 \dots^{2} - 1.26028 \dots + 1 = 2.67661 \dots

f^{^{'}} (u_{35}) = 6 \cdot 1.26028 \dots^{5} - 5 \cdot 1.26028 \dots^{4} + 4 \cdot 1.26028 \dots^{3} - 3 \cdot 1.26028 \dots^{2} + 2 \cdot 1.26028 \dots - 1 = 11.22518 \dots

u_{36} = 1.02183 \dots

Δ u_{36} = ∣ 1.02183 \dots - 1.26028 \dots ∣ = 0.23844 \dots

Δ u_{36} = 0.23844 \dots

u_{37} = 0.70827 \dots : Δ u_{37} = 0.31356 \dots

f (u_{36}) = 1.02183 \dots^{6} - 1.02183 \dots^{5} + 1.02183 \dots^{4} - 1.02183 \dots^{3} + 1.02183 \dots^{2} - 1.02183 \dots + 1 = 1.06994 \dots

f^{^{'}} (u_{36}) = 6 \cdot 1.02183 \dots^{5} - 5 \cdot 1.02183 \dots^{4} + 4 \cdot 1.02183 \dots^{3} - 3 \cdot 1.02183 \dots^{2} + 2 \cdot 1.02183 \dots - 1 = 3.41216 \dots

u_{37} = 0.70827 \dots

Δ u_{37} = ∣ 0.70827 \dots - 1.02183 \dots ∣ = 0.31356 \dots

Δ u_{37} = 0.31356 \dots

Cannot find solution

The solution is

No Solution for u \in R

Solve

u^{14} - u^{7} + 1 = 0 :

No Solution for

u \in R

u^{14} - u^{7} + 1 = 0

Find one solution for

u^{14} - u^{7} + 1 = 0

using Newton-Raphson:

No Solution for

u \in R

u^{14} - u^{7} + 1 = 0

Newton-Raphson Approximation Definition

f (u) = u^{14} - u^{7} + 1

Find

f^{^{'}} (u) : 14 u^{13} - 7 u^{6}

\frac{d}{d u} (u^{14} - u^{7} + 1)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{14}) - \frac{d}{d u} (u^{7}) + \frac{d}{d u} (1)

\frac{d}{d u} (u^{14}) = 14 u^{13}

\frac{d}{d u} (u^{14})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 14 u^{14 - 1}

Simplify

= 14 u^{13}

\frac{d}{d u} (u^{7}) = 7 u^{6}

\frac{d}{d u} (u^{7})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 7 u^{7 - 1}

Simplify

= 7 u^{6}

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 14 u^{13} - 7 u^{6} + 0

Simplify

= 14 u^{13} - 7 u^{6}

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.85714 \dots : Δ u_{1} = 0.14285 \dots

f (u_{0}) = (- 1)^{14} - (- 1)^{7} + 1 = 3

f^{^{'}} (u_{0}) = 14 (- 1)^{13} - 7 (- 1)^{6} = - 21

u_{1} = - 0.85714 \dots

Δ u_{1} = ∣ - 0.85714 \dots - (- 1) ∣ = 0.14285 \dots

Δ u_{1} = 0.14285 \dots

u_{2} = - 0.54502 \dots : Δ u_{2} = 0.31211 \dots

f (u_{1}) = (- 0.85714 \dots)^{14} - (- 0.85714 \dots)^{7} + 1 = 1.45546 \dots

f^{^{'}} (u_{1}) = 14 (- 0.85714 \dots)^{13} - 7 (- 0.85714 \dots)^{6} = - 4.66319 \dots

u_{2} = - 0.54502 \dots

Δ u_{2} = ∣ - 0.54502 \dots - (- 0.85714 \dots) ∣ = 0.31211 \dots

Δ u_{2} = 0.31211 \dots

u_{3} = 4.83036 \dots : Δ u_{3} = 5.37539 \dots

f (u_{2}) = (- 0.54502 \dots)^{14} - (- 0.54502 \dots)^{7} + 1 = 1.01449 \dots

f^{^{'}} (u_{2}) = 14 (- 0.54502 \dots)^{13} - 7 (- 0.54502 \dots)^{6} = - 0.18872 \dots

u_{3} = 4.83036 \dots

Δ u_{3} = ∣ 4.83036 \dots - (- 0.54502 \dots) ∣ = 5.37539 \dots

Δ u_{3} = 5.37539 \dots

u_{4} = 4.48534 \dots : Δ u_{4} = 0.34502 \dots

f (u_{3}) = 4.83036 \dots^{14} - 4.83036 \dots^{7} + 1 = 3764539189.66291 \dots

f^{^{'}} (u_{3}) = 14 \cdot 4.83036 \dots^{13} - 7 \cdot 4.83036 \dots^{6} = 10910972868.24572

u_{4} = 4.48534 \dots

Δ u_{4} = ∣ 4.48534 \dots - 4.83036 \dots ∣ = 0.34502 \dots

Δ u_{4} = 0.34502 \dots

u_{5} = 4.16496 \dots : Δ u_{5} = 0.32037 \dots

f (u_{4}) = 4.48534 \dots^{14} - 4.48534 \dots^{7} + 1 = 1333906086.09062 \dots

f^{^{'}} (u_{4}) = 14 \cdot 4.48534 \dots^{13} - 7 \cdot 4.48534 \dots^{6} = 4163549426.74544 \dots

u_{5} = 4.16496 \dots

Δ u_{5} = ∣ 4.16496 \dots - 4.48534 \dots ∣ = 0.32037 \dots

Δ u_{5} = 0.32037 \dots

u_{6} = 3.86747 \dots : Δ u_{6} = 0.29749 \dots

f (u_{5}) = 4.16496 \dots^{14} - 4.16496 \dots^{7} + 1 = 472648196.17869 \dots

f^{^{'}} (u_{5}) = 14 \cdot 4.16496 \dots^{13} - 7 \cdot 4.16496 \dots^{6} = 1588783582.76017 \dots

u_{6} = 3.86747 \dots

Δ u_{6} = ∣ 3.86747 \dots - 4.16496 \dots ∣ = 0.29749 \dots

Δ u_{6} = 0.29749 \dots

u_{7} = 3.59123 \dots : Δ u_{7} = 0.27623 \dots

f (u_{6}) = 3.86747 \dots^{14} - 3.86747 \dots^{7} + 1 = 167474855.60144 \dots

f^{^{'}} (u_{6}) = 14 \cdot 3.86747 \dots^{13} - 7 \cdot 3.86747 \dots^{6} = 606271364.08925 \dots

u_{7} = 3.59123 \dots

Δ u_{7} = ∣ 3.59123 \dots - 3.86747 \dots ∣ = 0.27623 \dots

Δ u_{7} = 0.27623 \dots

u_{8} = 3.33473 \dots : Δ u_{8} = 0.25650 \dots

f (u_{7}) = 3.59123 \dots^{14} - 3.59123 \dots^{7} + 1 = 59341606.39963 \dots

f^{^{'}} (u_{7}) = 14 \cdot 3.59123 \dots^{13} - 7 \cdot 3.59123 \dots^{6} = 231351084.81736 \dots

u_{8} = 3.33473 \dots

Δ u_{8} = ∣ 3.33473 \dots - 3.59123 \dots ∣ = 0.25650 \dots

Δ u_{8} = 0.25650 \dots

u_{9} = 3.09656 \dots : Δ u_{9} = 0.23816 \dots

f (u_{8}) = 3.33473 \dots^{14} - 3.33473 \dots^{7} + 1 = 21026440.56959 \dots

f^{^{'}} (u_{8}) = 14 \cdot 3.33473 \dots^{13} - 7 \cdot 3.33473 \dots^{6} = 88283526.43084 \dots

u_{9} = 3.09656 \dots

Δ u_{9} = ∣ 3.09656 \dots - 3.33473 \dots ∣ = 0.23816 \dots

Δ u_{9} = 0.23816 \dots

u_{10} = 2.87542 \dots : Δ u_{10} = 0.22114 \dots

f (u_{9}) = 3.09656 \dots^{14} - 3.09656 \dots^{7} + 1 = 7450180.69725 \dots

f^{^{'}} (u_{9}) = 14 \cdot 3.09656 \dots^{13} - 7 \cdot 3.09656 \dots^{6} = 33689450.55443 \dots

u_{10} = 2.87542 \dots

Δ u_{10} = ∣ 2.87542 \dots - 3.09656 \dots ∣ = 0.22114 \dots

Δ u_{10} = 0.22114 \dots

u_{11} = 2.67009 \dots : Δ u_{11} = 0.20532 \dots

f (u_{10}) = 2.87542 \dots^{14} - 2.87542 \dots^{7} + 1 = 2639725.48192 \dots

f^{^{'}} (u_{10}) = 14 \cdot 2.87542 \dots^{13} - 7 \cdot 2.87542 \dots^{6} = 12856372.82329 \dots

u_{11} = 2.67009 \dots

Δ u_{11} = ∣ 2.67009 \dots - 2.87542 \dots ∣ = 0.20532 \dots

Δ u_{11} = 0.20532 \dots

u_{12} = 2.47947 \dots : Δ u_{12} = 0.19062 \dots

f (u_{11}) = 2.67009 \dots^{14} - 2.67009 \dots^{7} + 1 = 935266.72285 \dots

f^{^{'}} (u_{11}) = 14 \cdot 2.67009 \dots^{13} - 7 \cdot 2.67009 \dots^{6} = 4906369.06001 \dots

u_{12} = 2.47947 \dots

Δ u_{12} = ∣ 2.47947 \dots - 2.67009 \dots ∣ = 0.19062 \dots

Δ u_{12} = 0.19062 \dots

u_{13} = 2.30252 \dots : Δ u_{13} = 0.17695 \dots

f (u_{12}) = 2.47947 \dots^{14} - 2.47947 \dots^{7} + 1 = 331349.76638 \dots

f^{^{'}} (u_{12}) = 14 \cdot 2.47947 \dots^{13} - 7 \cdot 2.47947 \dots^{6} = 1872538.71063 \dots

u_{13} = 2.30252 \dots

Δ u_{13} = ∣ 2.30252 \dots - 2.47947 \dots ∣ = 0.17695 \dots

Δ u_{13} = 0.17695 \dots

u_{14} = 2.13829 \dots : Δ u_{14} = 0.16422 \dots

f (u_{13}) = 2.30252 \dots^{14} - 2.30252 \dots^{7} + 1 = 117380.28802 \dots

f^{^{'}} (u_{13}) = 14 \cdot 2.30252 \dots^{13} - 7 \cdot 2.30252 \dots^{6} = 714742.41872 \dots

u_{14} = 2.13829 \dots

Δ u_{14} = ∣ 2.13829 \dots - 2.30252 \dots ∣ = 0.16422 \dots

Δ u_{14} = 0.16422 \dots

u_{15} = 1.98593 \dots : Δ u_{15} = 0.15236 \dots

f (u_{14}) = 2.13829 \dots^{14} - 2.13829 \dots^{7} + 1 = 41575.02774 \dots

f^{^{'}} (u_{14}) = 14 \cdot 2.13829 \dots^{13} - 7 \cdot 2.13829 \dots^{6} = 272865.36981 \dots

u_{15} = 1.98593 \dots

Δ u_{15} = ∣ 1.98593 \dots - 2.13829 \dots ∣ = 0.15236 \dots

Δ u_{15} = 0.15236 \dots

u_{16} = 1.84465 \dots : Δ u_{16} = 0.14127 \dots

f (u_{15}) = 1.98593 \dots^{14} - 1.98593 \dots^{7} + 1 = 14721.50063 \dots

f^{^{'}} (u_{15}) = 14 \cdot 1.98593 \dots^{13} - 7 \cdot 1.98593 \dots^{6} = 104202.85485 \dots

u_{16} = 1.84465 \dots

Δ u_{16} = ∣ 1.84465 \dots - 1.98593 \dots ∣ = 0.14127 \dots

Δ u_{16} = 0.14127 \dots

u_{17} = 1.71378 \dots : Δ u_{17} = 0.13087 \dots

f (u_{16}) = 1.84465 \dots^{14} - 1.84465 \dots^{7} + 1 = 5210.48671 \dots

f^{^{'}} (u_{16}) = 14 \cdot 1.84465 \dots^{13} - 7 \cdot 1.84465 \dots^{6} = 39813.17132 \dots

u_{17} = 1.71378 \dots

Δ u_{17} = ∣ 1.71378 \dots - 1.84465 \dots ∣ = 0.13087 \dots

Δ u_{17} = 0.13087 \dots

u_{18} = 1.59272 \dots : Δ u_{18} = 0.12105 \dots

f (u_{17}) = 1.71378 \dots^{14} - 1.71378 \dots^{7} + 1 = 1842.86223 \dots

f^{^{'}} (u_{17}) = 14 \cdot 1.71378 \dots^{13} - 7 \cdot 1.71378 \dots^{6} = 15223.65016 \dots

u_{18} = 1.59272 \dots

Δ u_{18} = ∣ 1.59272 \dots - 1.71378 \dots ∣ = 0.12105 \dots

Δ u_{18} = 0.12105 \dots

u_{19} = 1.48102 \dots : Δ u_{19} = 0.11170 \dots

f (u_{18}) = 1.59272 \dots^{14} - 1.59272 \dots^{7} + 1 = 651.06020 \dots

f^{^{'}} (u_{18}) = 14 \cdot 1.59272 \dots^{13} - 7 \cdot 1.59272 \dots^{6} = 5828.26805 \dots

u_{19} = 1.48102 \dots

Δ u_{19} = ∣ 1.48102 \dots - 1.59272 \dots ∣ = 0.11170 \dots

Δ u_{19} = 0.11170 \dots

u_{20} = 1.37828 \dots : Δ u_{20} = 0.10273 \dots

f (u_{19}) = 1.48102 \dots^{14} - 1.48102 \dots^{7} + 1 = 229.63507 \dots

f^{^{'}} (u_{19}) = 14 \cdot 1.48102 \dots^{13} - 7 \cdot 1.48102 \dots^{6} = 2235.14206 \dots

u_{20} = 1.37828 \dots

Δ u_{20} = ∣ 1.37828 \dots - 1.48102 \dots ∣ = 0.10273 \dots

Δ u_{20} = 0.10273 \dots

u_{21} = 1.28417 \dots : Δ u_{21} = 0.09411 \dots

f (u_{20}) = 1.37828 \dots^{14} - 1.37828 \dots^{7} + 1 = 80.82807 \dots

f^{^{'}} (u_{20}) = 14 \cdot 1.37828 \dots^{13} - 7 \cdot 1.37828 \dots^{6} = 858.84639 \dots

u_{21} = 1.28417 \dots

Δ u_{21} = ∣ 1.28417 \dots - 1.37828 \dots ∣ = 0.09411 \dots

Δ u_{21} = 0.09411 \dots

u_{22} = 1.19813 \dots : Δ u_{22} = 0.08603 \dots

f (u_{21}) = 1.28417 \dots^{14} - 1.28417 \dots^{7} + 1 = 28.40880 \dots

f^{^{'}} (u_{21}) = 14 \cdot 1.28417 \dots^{13} - 7 \cdot 1.28417 \dots^{6} = 330.20328 \dots

u_{22} = 1.19813 \dots

Δ u_{22} = ∣ 1.19813 \dots - 1.28417 \dots ∣ = 0.08603 \dots

Δ u_{22} = 0.08603 \dots

u_{23} = 1.11867 \dots : Δ u_{23} = 0.07945 \dots

f (u_{22}) = 1.19813 \dots^{14} - 1.19813 \dots^{7} + 1 = 10.01843 \dots

f^{^{'}} (u_{22}) = 14 \cdot 1.19813 \dots^{13} - 7 \cdot 1.19813 \dots^{6} = 126.08661 \dots

u_{23} = 1.11867 \dots

Δ u_{23} = ∣ 1.11867 \dots - 1.19813 \dots ∣ = 0.07945 \dots

Δ u_{23} = 0.07945 \dots

u_{24} = 1.04084 \dots : Δ u_{24} = 0.07783 \dots

f (u_{23}) = 1.11867 \dots^{14} - 1.11867 \dots^{7} + 1 = 3.61456 \dots

f^{^{'}} (u_{23}) = 14 \cdot 1.11867 \dots^{13} - 7 \cdot 1.11867 \dots^{6} = 46.43999 \dots

u_{24} = 1.04084 \dots

Δ u_{24} = ∣ 1.04084 \dots - 1.11867 \dots ∣ = 0.07783 \dots

Δ u_{24} = 0.07783 \dots

u_{25} = 0.94342 \dots : Δ u_{25} = 0.09742 \dots

f (u_{24}) = 1.04084 \dots^{14} - 1.04084 \dots^{7} + 1 = 1.42806 \dots

f^{^{'}} (u_{24}) = 14 \cdot 1.04084 \dots^{13} - 7 \cdot 1.04084 \dots^{6} = 14.65834 \dots

u_{25} = 0.94342 \dots

Δ u_{25} = ∣ 0.94342 \dots - 1.04084 \dots ∣ = 0.09742 \dots

Δ u_{25} = 0.09742 \dots

u_{26} = 0.46673 \dots : Δ u_{26} = 0.47668 \dots

f (u_{25}) = 0.94342 \dots^{14} - 0.94342 \dots^{7} + 1 = 0.77728 \dots

f^{^{'}} (u_{25}) = 14 \cdot 0.94342 \dots^{13} - 7 \cdot 0.94342 \dots^{6} = 1.63060 \dots

u_{26} = 0.46673 \dots

Δ u_{26} = ∣ 0.46673 \dots - 0.94342 \dots ∣ = 0.47668 \dots

Δ u_{26} = 0.47668 \dots

Cannot find solution

The solution is

No Solution for u \in R

The solutions are

u = 0, u = - 1

Substitute back

u = cos (x)

cos (x) = 0, cos (x) = - 1

cos (x) = 0, cos (x) = - 1

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

General solutions for

cos (x) = - 1

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = π + 2 πn

Popular Examples

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4(cos(x)+1)cos(x)=3

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sin^{2} (x) + 1 = cos^{2} (x) - 2 sin^{4} (x)

Frequently Asked Questions (FAQ)

What is the general solution for cos^{23}(x)+cos^2(x)=0 ?
The general solution for cos^{23}(x)+cos^2(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin,x=pi+2pin