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

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 i2a−i2c=10 Solve: i2a−i2c=10 and a+c=0 We get a=−10i and c=10i Thus, the quadratic factors are: (z2−10iz−i2) and (z2+10iz

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Kye

Answered question

2021-09-13

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

Answer & Explanation

Jozlyn

Skilled2021-09-14Added 85 answers

Solution:
Given:

z^{4} + \sqrt{10} z + \frac{1}{4}

Let the two quadratic factors be

(z^{2} + a z + b)

and

(z^{2} + c z + d)

Then,

z^{4} + \sqrt{10} z + \frac{1}{4} = (z^{2} + a z + b) (z^{2} + c z + d)

z^{4} + \sqrt{10} z + \frac{1}{4} = z^{4} + (a + c) z^{3} + (b + d) z^{2} + (b c + a d) z + b d

Compare the coefficients of power of z on both sides

a + c = 0

b + d = 0

b c + a d = \sqrt{10}

b d = \frac{1}{4}

Now,

b d = \frac{1}{4}

gives

b = \frac{1}{4 d}

, substitute

b = \frac{1}{4 d}

into

b + d = 0

\frac{1}{4 d} + d = 0

\frac{1 + 4 d^{2}}{4 d} = 0

which gives

4 d^{2} = - 1

d = \pm \frac{i}{2}

Take

d = \frac{i}{2}

, substitute this into

b = \frac{1}{4 d}

b = \frac{1}{2 i}

b = - \frac{i^{2}}{2 i}

b = - \frac{i}{2}

Now, substitute

d = \frac{i}{2}

and

b = - \frac{i}{2}

into

b c + a d = \sqrt{10}

, we get

\frac{i}{2} a - \frac{i}{2} c = \sqrt{10}

Solve:

\frac{i}{2} a - \frac{i}{2} c = \sqrt{10}

and a+c=0
We get

a = - \sqrt{10} i

and

c = \sqrt{10} i

Thus, the quadratic factors are:

(z^{2} - \sqrt{10} i z - \frac{i}{2})

and

(z^{2} + \sqrt{10} i z

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