Question

Solve the polynomial equationx^4 - 5x^2 -10x -6 =0 given that -1 + i is a root of the equation

Answer 1

we have the polynomial

x^4 - 5x^2 -10x -6 =0

If (-1+i) is a root

then

by the conjugate theorem

(-1-i) is also a root

The factors of the given roots are

[x-(-1+i)] and [x-(-1-i)]

step 1

Multiply the factors

[x-(-1+i)][x-(-1-i)]=x^2-x(-1-i)-x(-1+i)+(-1+i)(-1-i)

=x^2+x+xi+x-xi+(-1)^2-(i)^2

=x^2+2x+1-i^2

Remember that i^2=-1

=x^2+2x+1-(-1)

=x^2+2x+2

Step 2

Divide the given polynomial by the quadratic equation (x^2+2x+2)

x^4 - 5x^2 -10x -6 : (x^2+2x+2)

x^2-2x-3

-x^4-2x^3-2x^2

--------------------------

-2x^3-7x^2-10x-6

2x^3+4x^2+4x

------------------------

-3x^2-6x-6

3x^2+6x+6

--------------------

0

so

we have that

x^4 - 5x^2 -10x -6=(x^2+2x+2)(x^2-2x-3)

step 3

Solve the quadratic equation

(x^2-2x-3)=0

Using the formula

a=1

b=-2

c=-3

substitute

$x=\frac{-(-2)\pm\sqrt{-2^2-4(1)(-3)}}{2(1)}$$x=\frac{2\pm4}{2}$

therefore

The roots of the given polynomial are

$\begin{gathered} x=-1+i \\ x=-1-i \\ \\ x=3 \\ \\ x=-1 \end{gathered}$