Question

Given that (X+3) is a factor of the expression x^3 + 4x^2 + px + 3 , find the value of p. Hence, solve the equation x^3 + 4x^2 + px + 3=0, expressing the complex number in the form a + bi​

Answer 1

Answer:

$p=4$

$x=\frac{-1}{2} \pm \frac{\sqrt{3}}{2}i$

Step-by-step explanation:

We are given (x+3) is a factor of $x^3+4x^2+px+3$, which means if were to plug in -3, the result is 0.

Let's write that down:

$(-3)^3+4(-3)^2+p(-3)+3=0$

$-27+36-3p+3=0$

$9-3p+3=0$

$9+3-3p=0$

$12-3p=0$

$12=3p$

$p=4$

So the cubic equation is actually $x^3+4x^2+4x+3=0$ that they wish we solve for $x$.

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 $x^3+4x^2+4x+3$ is $(x^2+x+1)$.

We must solve $x^2+x+1=0$.

Compare this to $ax^2+bx+c=0$.

We have $a=1,b=1, \text{ and } c=1$.

The quadratic formula is

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

Plug in the numbers we have for $a,b, \text{ and } c$.

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

Simplify inside the square root while also performing the one operation on bottom:

$x=\frac{-1 \pm \sqrt{1-4}}{2}$

$x=\frac{-1 \pm \sqrt{-3}}{2}$

Now our answer will include an imaginary part because of that sqrt(negative number).

The imaginary unit is $i=\sqrt{-1}$.

So our final answer is:

$x=\frac{-1}{2} \pm \frac{\sqrt{3}}{2}i$