How do you find all solutions to ^{3}+1=0?

taoigas

taoigas

Answered question

2022-01-31

How do you find all solutions to {3}+1=0?

Answer & Explanation

nebajcioz

nebajcioz

Beginner2022-02-01Added 15 answers

Step 1
Using synthetic division and the fact that x=1 is obviously a solution we find that we can expand this to:
(x+1)(x2x+1)=0
In order to have LHS=RHS need one of the brackets to be equal to zero, ie
1) (x+1)=0
2) (x2x+1)=0
Fro m1 we note that x=1 is a solution. We shall solve 2 using the quadratic formula:
x2x+1=0
x=1±(1)24(1)(1)2=1±32=1±3i2
Rosa Nicholson

Rosa Nicholson

Beginner2022-02-02Added 13 answers

Step 1
Given: x3+1=0
By Rational Root Theorem, all rational roots of a polynomial are in the form pq, where p divides the constant term 1 and q divides the leading coefficient 1. List all candidates pq
±1
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x=1
By Factor theorem, xk is a factor of the polynomial for each root k. Divide x3+1 by z+1 to get x2x+1. Solve the equation where the result equals to 0.
x2x+1=0
All equations of the form ax2+bx+c=0 can be solved using the quadratic formula:
b±b24ac2a
Substitute 1 for a, -1 for b, and 1 for c in the quadratic formula.
x=(1)±(1)24×1×12
Do the calculations.
x=1±32
Since the square root of a negative number is not defined in the real field, there are no solutions.
x0
List all found solutions.
x=1

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