Find all roots of each of the of the following quartic polynomials:a) z^4+\sqrt{10}z+\frac{1}{4} using factorization into two quadratic polynomials

Kye

Kye

Answered question

2021-09-13

Find all roots of each of the of the following quartic polynomials:
a) z4+10z+14 using factorization into two quadratic polynomials

Answer & Explanation

Jozlyn

Jozlyn

Skilled2021-09-14Added 85 answers

Solution:
Given:
z4+10z+14
Let the two quadratic factors be (z2+az+b) and (z2+cz+d)
Then,
z4+10z+14=(z2+az+b)(z2+cz+d)
z4+10z+14=z4+(a+c)z3+(b+d)z2+(bc+ad)z+bd
Compare the coefficients of power of z on both sides
a+c=0
b+d=0
bc+ad=10
bd=14
Now,
bd=14 gives b=14d, substitute b=14d into b+d=0
14d+d=0
1+4d24d=0
which gives
4d2=1
d=±i2
Take d=i2, substitute this into b=14d
b=12i
b=i22i
b=i2
Now, substitute d=i2 and b=i2 into bc+ad=10, we get
i2ai2c=10
Solve: i2ai2c=10 and a+c=0
We get a=10i and c=10i
Thus, the quadratic factors are:
(z210izi2) and (z2+10iz

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?