Question

Factor the expression completely over the complex numbers. X3+5x2+9x+45

Answer 1

$x^3+5x^2+9x+45=x^2(x+5)+9(x+5)=(x+5)(x^2+9)=(*)\\\\x^2+9=x^2+3^2=x^2-(-1)(3^2)=x^2-(i^2)(3^2)=x^2-(3i)^2\\\\i=\sqrt{-1}\to i^2=-1\\\\(*)=(x+5)[x^2-(3i)^2]=(x+5)(x-3i)(x+3i)\\\\Used:\ a^2-b^2=(a-b)(a+b)\\\\Answer:\ x^3+5x^2+9x+45=(x+5)(x-3i)(x+3i)$

Answer 2

Answer:

The expression after completely factorizing is given by:

        $x^3+5x^2+9x+45=(x+5)(x+3i)(x-3i)$

Step-by-step explanation:

We are given an algebraic expression in terms of the variable x as follows:

                 $x^3+5x^2+9x+45$

We are asked to factor the expression completely.

i.e. the expression is written as follows:

  $x^3+5x^2+9x+45=x^2(x+5)+9(x+5)$

which is nothing but:

  $x^3+5x^2+9x+45=(x^2+9)(x+5)$

Now, we know that any quadratic equation of the type:

                  $ax^2+bx+c=0$

has solution by the quadratic formula as:

$x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$

Here we have to find the solution of the quadratic equation:

$x^2+9=0$

i.e.

$a=1,\ b=0\ and\ c=9$

Hence, the solution is given by:

$x=\dfrac{-0\pm \sqrt{0^2-4\times 1\times 9}}{2\times 1}\\\\x=\dfrac{\pm \sqrt{-36}}{2}\\\\x=\dfrac{\pm 6i}{2}\\\\x=\pm 3i\\\\x=3i\ and\ x=-3i\\\\x-3i=0\ and\ x+3i=0$

Hence, we have the expression as follows:

     $x^3+5x^2+9x+45=(x+5)(x+3i)(x-3i)$