Respuesta :
Answer:
[tex]p=4[/tex]
[tex]x=\frac{-1}{2} \pm \frac{\sqrt{3}}{2}i[/tex]
Step-by-step explanation:
We are given (x+3) is a factor of [tex]x^3+4x^2+px+3[/tex], which means if were to plug in -3, the result is 0.
Let's write that down:
[tex](-3)^3+4(-3)^2+p(-3)+3=0[/tex]
[tex]-27+36-3p+3=0[/tex]
[tex]9-3p+3=0[/tex]
[tex]9+3-3p=0[/tex]
[tex]12-3p=0[/tex]
[tex]12=3p[/tex]
[tex]p=4[/tex]
So the cubic equation is actually [tex]x^3+4x^2+4x+3=0[/tex] that they wish we solve for [tex]x[/tex].
To find another factor of the given cubic expression on the left, I'm going to use synthetic division with that polynomial and (x+3) where (x+3) is divisor. Since (x+3) is the divisor, -3 will be on the outside like so:
-3 | 1 4 4 3
| -3 -3 -3
---------------------
1 1 1 0
So the other factor of [tex]x^3+4x^2+4x+3[/tex] is [tex](x^2+x+1)[/tex].
We must solve [tex]x^2+x+1=0[/tex].
Compare this to [tex]ax^2+bx+c=0[/tex].
We have [tex]a=1,b=1, \text{ and } c=1[/tex].
The quadratic formula is
[tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex].
Plug in the numbers we have for [tex]a,b, \text{ and } c[/tex].
[tex]x=\frac{-1 \pm \sqrt{1^2-4(1)(1)}}{2(1)}[/tex].
Simplify inside the square root while also performing the one operation on bottom:
[tex]x=\frac{-1 \pm \sqrt{1-4}}{2}[/tex]
[tex]x=\frac{-1 \pm \sqrt{-3}}{2}[/tex]
Now our answer will include an imaginary part because of that sqrt(negative number).
The imaginary unit is [tex]i=\sqrt{-1}[/tex].
So our final answer is:
[tex]x=\frac{-1}{2} \pm \frac{\sqrt{3}}{2}i[/tex]
