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2csc^3(x)+21csc^2(x)+55csc(x)+42=0

Solution

2 csc^{3} (x) + 21 csc^{2} (x) + 55 csc (x) + 42 = 0

Solution

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn, x = - 0.14334 \dots + 2 πn, x = π + 0.14334 \dots + 2 πn

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = - 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 221.81031 \dots^{\circ} + 36 0^{\circ} n, x = - 8.21321 \dots^{\circ} + 36 0^{\circ} n, x = 188.21321 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 csc^{3} (x) + 21 csc^{2} (x) + 55 csc (x) + 42 = 0

Solve by substitution

2 csc^{3} (x) + 21 csc^{2} (x) + 55 csc (x) + 42 = 0

Let:

csc (x) = u

2 u^{3} + 21 u^{2} + 55 u + 42 = 0

2 u^{3} + 21 u^{2} + 55 u + 42 = 0 : u = - 2, u = - \frac{3}{2}, u = - 7

2 u^{3} + 21 u^{2} + 55 u + 42 = 0

Factor

2 u^{3} + 21 u^{2} + 55 u + 42 : (u + 2) (2 u + 3) (u + 7)

2 u^{3} + 21 u^{2} + 55 u + 42

Use the rational root theorem

a_{0} = 42, a_{n} = 2

The dividers of

a_{0} : 1, 2, 3, 6, 7, 14, 21, 42,

The dividers of

a_{n} : 1, 2

Therefore, check the following rational numbers:

\pm \frac{1 , 2 , 3 , 6 , 7 , 14 , 21 , 42}{1 , 2}

- \frac{2}{1}

is a root of the expression, so factor out

u + 2

= (u + 2) \frac{2 u ^{3} + 21 u ^{2} + 55 u + 42}{u + 2}

\frac{2 u ^{3} + 21 u ^{2} + 55 u + 42}{u + 2} = 2 u^{2} + 17 u + 21

\frac{2 u ^{3} + 21 u ^{2} + 55 u + 42}{u + 2}

Divide

\frac{2 u ^{3} + 21 u ^{2} + 55 u + 42}{u + 2} : \frac{2 u ^{3} + 21 u ^{2} + 55 u + 42}{u + 2} = 2 u^{2} + \frac{17 u ^{2} + 55 u + 42}{u + 2}

Divide the leading coefficients of the numerator

2 u^{3} + 21 u^{2} + 55 u + 42

and the divisor

u + 2 : \frac{2 u ^{3}}{u} = 2 u^{2}

Quotient = 2 u^{2}

Multiply

u + 2

2 u^{2} : 2 u^{3} + 4 u^{2}

Subtract

2 u^{3} + 4 u^{2}

from

2 u^{3} + 21 u^{2} + 55 u + 42

to get new remainder

Remainder = 17 u^{2} + 55 u + 42

Therefore

\frac{2 u ^{3} + 21 u ^{2} + 55 u + 42}{u + 2} = 2 u^{2} + \frac{17 u ^{2} + 55 u + 42}{u + 2}

= 2 u^{2} + \frac{17 u ^{2} + 55 u + 42}{u + 2}

Divide

\frac{17 u ^{2} + 55 u + 42}{u + 2} : \frac{17 u ^{2} + 55 u + 42}{u + 2} = 17 u + \frac{21 u + 42}{u + 2}

Divide the leading coefficients of the numerator

17 u^{2} + 55 u + 42

and the divisor

u + 2 : \frac{17 u ^{2}}{u} = 17 u

Quotient = 17 u

Multiply

u + 2

17 u : 17 u^{2} + 34 u

Subtract

17 u^{2} + 34 u

from

17 u^{2} + 55 u + 42

to get new remainder

Remainder = 21 u + 42

Therefore

\frac{17 u ^{2} + 55 u + 42}{u + 2} = 17 u + \frac{21 u + 42}{u + 2}

= 2 u^{2} + 17 u + \frac{21 u + 42}{u + 2}

Divide

\frac{21 u + 42}{u + 2} : \frac{21 u + 42}{u + 2} = 21

Divide the leading coefficients of the numerator

21 u + 42

and the divisor

u + 2 : \frac{21 u}{u} = 21

Quotient = 21

Multiply

u + 2

21 : 21 u + 42

Subtract

21 u + 42

from

21 u + 42

to get new remainder

Remainder = 0

Therefore

\frac{21 u + 42}{u + 2} = 21

= 2 u^{2} + 17 u + 21

= 2 u^{2} + 17 u + 21

Factor

2 u^{2} + 17 u + 21 : (2 u + 3) (u + 7)

2 u^{2} + 17 u + 21

Break the expression into groups

2 u^{2} + 17 u + 21

Definition

Factors of

42 : 1, 2, 3, 6, 7, 14, 21, 42

42

Divisors (Factors)

Find the Prime factors of

42 : 2, 3, 7

42

42

divides by

242 = 21 \cdot 2

= 2 \cdot 21

21

divides by

321 = 7 \cdot 3

= 2 \cdot 3 \cdot 7

2, 3, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 7

Multiply the prime factors of

42 : 6, 14, 21

2 \cdot 3 = 6

2 \cdot 7 = 14

6, 14, 21

6, 14, 21

Add the prime factors:

2, 3, 7

Add 1 and the number

42

itself

1, 42

The factors of

42

1, 2, 3, 6, 7, 14, 21, 42

For every two factors such that

u * v = 42,

check if

u + v = 17

Check

u = 1, v = 42 : u * v = 42, u + v = 43 \Rightarrow

FalseCheck

u = 2, v = 21 : u * v = 42, u + v = 23 \Rightarrow

False

u = 3, v = 14

Group into

(a x^{2} + ux) + (vx + c)

(2 u^{2} + 3 u) + (14 u + 21)

= (2 u^{2} + 3 u) + (14 u + 21)

Factor out

u

from

2 u^{2} + 3 u : u (2 u + 3)

2 u^{2} + 3 u

Apply exponent rule:

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

u^{2} = uu

= 2 uu + 3 u

Factor out common term

u

= u (2 u + 3)

Factor out

7

from

14 u + 21 : 7 (2 u + 3)

14 u + 21

Rewrite

21

7 \cdot 3

Rewrite

14

7 \cdot 2

= 7 \cdot 2 u + 7 \cdot 3

Factor out common term

7

= 7 (2 u + 3)

= u (2 u + 3) + 7 (2 u + 3)

Factor out common term

2 u + 3

= (2 u + 3) (u + 7)

= (u + 2) (2 u + 3) (u + 7)

(u + 2) (2 u + 3) (u + 7) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u + 2 = 0 or 2 u + 3 = 0 or u + 7 = 0

Solve

u + 2 = 0 : u = - 2

u + 2 = 0

Move

2

to the right side

u + 2 = 0

Subtract

2

from both sides

u + 2 - 2 = 0 - 2

Simplify

u = - 2

u = - 2

Solve

2 u + 3 = 0 : u = - \frac{3}{2}

2 u + 3 = 0

Move

3

to the right side

2 u + 3 = 0

Subtract

3

from both sides

2 u + 3 - 3 = 0 - 3

Simplify

2 u = - 3

2 u = - 3

Divide both sides by

2

2 u = - 3

Divide both sides by

2

\frac{2 u}{2} = \frac{- 3}{2}

Simplify

u = - \frac{3}{2}

u = - \frac{3}{2}

Solve

u + 7 = 0 : u = - 7

u + 7 = 0

Move

7

to the right side

u + 7 = 0

Subtract

7

from both sides

u + 7 - 7 = 0 - 7

Simplify

u = - 7

u = - 7

The solutions are

u = - 2, u = - \frac{3}{2}, u = - 7

Substitute back

u = csc (x)

csc (x) = - 2, csc (x) = - \frac{3}{2}, csc (x) = - 7

csc (x) = - 2, csc (x) = - \frac{3}{2}, csc (x) = - 7

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = - 2

General solutions for

csc (x) = - 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = - \frac{3}{2} : x = arccsc (- \frac{3}{2}) + 2 πn, x = π + arccsc (\frac{3}{2}) + 2 πn

csc (x) = - \frac{3}{2}

Apply trig inverse properties

csc (x) = - \frac{3}{2}

General solutions for

csc (x) = - \frac{3}{2}

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

x = arccsc (- \frac{3}{2}) + 2 πn, x = π + arccsc (\frac{3}{2}) + 2 πn

x = arccsc (- \frac{3}{2}) + 2 πn, x = π + arccsc (\frac{3}{2}) + 2 πn

csc (x) = - 7 : x = arccsc (- 7) + 2 πn, x = π + arccsc (7) + 2 πn

csc (x) = - 7

Apply trig inverse properties

csc (x) = - 7

General solutions for

csc (x) = - 7

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

x = arccsc (- 7) + 2 πn, x = π + arccsc (7) + 2 πn

x = arccsc (- 7) + 2 πn, x = π + arccsc (7) + 2 πn

Combine all the solutions

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = arccsc (- \frac{3}{2}) + 2 πn, x = π + arccsc (\frac{3}{2}) + 2 πn, x = arccsc (- 7) + 2 πn, x = π + arccsc (7) + 2 πn

Show solutions in decimal form

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn, x = - 0.14334 \dots + 2 πn, x = π + 0.14334 \dots + 2 πn

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