z4+z3+z2+3z-6=0

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.

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.

z-1=0

Add 1 to both sides of the equation.

z=1

z=1

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