# Solve for z z^4+z^3+z^2+3z-6=0 z4+z3+z2+3z-6=0
Factor the left side of the equation.
Regroup terms.
z3+3z+z4+z2-6=0
Factor z out of z3+3z.
Factor z out of z3.
z⋅z2+3z+z4+z2-6=0
Factor z out of 3z.
z⋅z2+z⋅3+z4+z2-6=0
Factor z out of z⋅z2+z⋅3.
z(z2+3)+z4+z2-6=0
z(z2+3)+z4+z2-6=0
Rewrite z4 as (z2)2.
z(z2+3)+(z2)2+z2-6=0
Let u=z2. Substitute u for all occurrences of z2.
z(z2+3)+u2+u-6=0
Factor u2+u-6 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -6 and whose sum is 1.
-2,3
Write the factored form using these integers.
z(z2+3)+(u-2)(u+3)=0
z(z2+3)+(u-2)(u+3)=0
Replace all occurrences of u with z2.
z(z2+3)+(z2-2)(z2+3)=0
Factor z2+3 out of z(z2+3)+(z2-2)(z2+3).
Factor z2+3 out of z(z2+3).
(z2+3)z+(z2-2)(z2+3)=0
Factor z2+3 out of (z2-2)(z2+3).
(z2+3)z+(z2+3)(z2-2)=0
Factor z2+3 out of (z2+3)z+(z2+3)(z2-2).
(z2+3)(z+z2-2)=0
(z2+3)(z+z2-2)=0
Let u=z. Substitute u for all occurrences of z.
(z2+3)(u+u2-2)=0
Factor u+u2-2 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -2 and whose sum is 1.
-1,2
Write the factored form using these integers.
(z2+3)((u-1)(u+2))=0
(z2+3)((u-1)(u+2))=0
Factor.
Replace all occurrences of u with z.
(z2+3)((z-1)(z+2))=0
Remove unnecessary parentheses.
(z2+3)(z-1)(z+2)=0
(z2+3)(z-1)(z+2)=0
(z2+3)(z-1)(z+2)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
z2+3=0
z-1=0
z+2=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
z2+3=0
Subtract 3 from both sides of the equation.
z2=-3
Take the square root of both sides of the equation to eliminate the exponent on the left side.
z=±-3
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite -3 as -1(3).
z=±-1⋅3
Rewrite -1(3) as -1⋅3.
z=±-1⋅3
Rewrite -1 as i.
z=±i3
z=±i3
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the ± to find the first solution.
z=i3
Next, use the negative value of the ± to find the second solution.
z=-i3
The complete solution is the result of both the positive and negative portions of the solution.
z=i3,-i3
z=i3,-i3
z=i3,-i3
z=i3,-i3
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
z-1=0
Add 1 to both sides of the equation.
z=1
z=1
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
z+2=0
Subtract 2 from both sides of the equation.
z=-2
z=-2
The final solution is all the values that make (z2+3)(z-1)(z+2)=0 true.
z=i3,-i3,1,-2
Solve for z z^4+z^3+z^2+3z-6=0

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