Question

If f(x) = 2x^3 + 10x^2 + 18x + 10 and x + 1 is a factor of f(x), then find all of the zeros of f(x) algebraically

Answer 1

Given the polynomial:

`f(x)=2x^3+10x^2+18x+10`

We know that (x + 1) is a factor of f(x). We divide f(x) by (x + 1):

Then:

`f(x)=(x+1)(2x^2+8x+10)=2(x+1)(x^2+4x+5)`

For the quadratic term, we solve the following equation:

`x^2+4x+5=0`

Using the general solution for quadratic equations:

`\begin{gathered} x=(-4\pm√(4^2-4\cdot1\cdot5))/(2\cdot1)=(-4\pm√(16-20))/(2)=(-4\pm√(4))/(2) \\ \\ \therefore x=-2\pm i \end{gathered}`

The zeros of f(x) are:

`\begin{gathered} x_1=-1 \\ \\ x_2=-2-i \\ \\ x_3=-2+i \end{gathered}`