Algebra
Calculus
Trigonometry
Matrix
Question
Factor the expression
(z+1)2(z2−2z−1)
Evaluate
z4−4z2−4z−1
Calculate
z4−2z3−z2+2z3−4z2−2z+z2−2z−1
Rewrite the expression
z2×z2−z2×2z−z2+2z×z2−2z×2z−2z+z2−2z−1
Factor out z2 from the expression
z2(z2−2z−1)+2z×z2−2z×2z−2z+z2−2z−1
Factor out 2z from the expression
z2(z2−2z−1)+2z(z2−2z−1)+z2−2z−1
Factor out z2−2z−1 from the expression
(z2+2z+1)(z2−2z−1)
Solution
(z+1)2(z2−2z−1)
Show Solution
Find the roots
z1=−1,z2=1−2,z3=1+2
Alternative Form
z1=−1,z2≈−0.414214,z3≈2.414214
Evaluate
z4−4z2−4z−1
To find the roots of the expression,set the expression equal to 0
z4−4z2−4z−1=0
Factor the expression
(z+1)2(z2−2z−1)=0
Separate the equation into 2 possible cases
(z+1)2=0z2−2z−1=0
Solve the equation
More Steps
Evaluate
(z+1)2=0
The only way a power can be 0 is when the base equals 0
z+1=0
Move the constant to the right-hand side and change its sign
z=0−1
Removing 0 doesn't change the value,so remove it from the expression
z=−1
z=−1z2−2z−1=0
Solve the equation
More Steps
Evaluate
z2−2z−1=0
Substitute a=1,b=−2 and c=−1 into the quadratic formula z=2a−b±b2−4ac
z=22±(−2)2−4(−1)
Simplify the expression
More Steps
Evaluate
(−2)2−4(−1)
Simplify
(−2)2−(−4)
Rewrite the expression
22−(−4)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
22+4
Evaluate the power
4+4
Add the numbers
8
z=22±8
Simplify the radical expression
More Steps
Evaluate
8
Write the expression as a product where the root of one of the factors can be evaluated
4×2
Write the number in exponential form with the base of 2
22×2
The root of a product is equal to the product of the roots of each factor
22×2
Reduce the index of the radical and exponent with 2
22
z=22±22
Separate the equation into 2 possible cases
z=22+22z=22−22
Simplify the expression
z=1+2z=22−22
Simplify the expression
z=1+2z=1−2
z=−1z=1+2z=1−2
Solution
z1=−1,z2=1−2,z3=1+2
Alternative Form
z1=−1,z2≈−0.414214,z3≈2.414214
Show Solution