How to find zeros (x-intercepts)...?

Question

How do I find the zeros of f(x) = 2(x^3) + 6(x^2) - 4.5x - 13.5 WITHOUT using the calculator?

I attempted to use the possible rational zeros & synthetic division, but it's a bit complicated...

Step-by-step detailed instructions would be helpful :)

Answer 1

2x^2(x+3) - 4.5(x+3)
(2x^2 -4.5)(x+3)

x = -3 or x^2 = 2.25; x = +/-1.5

Answer 2

You need to find zeros, so you let 0 be f(x) as that is the meaning of zeros.
Multiply by 2 each side and get:
4x^3+12x^2-9x-27=0
Use rational zeros.
p/q=+-1,+-3,+-9...
and many others.
Try 1 and -1. It doesn't work. Try -3 and 3. 3 doesn't but -3 does!
Therefore one factor is x+3.
Divide the function by x+3.
You get:
4x^2-9=0
(2x-3)(2x+3)=0
Thus the zeros are x=-3, x=3/2, and x=-3/2
However, if you write the function in factors, you have:
f(x)=(x+3)(2x-3)(2x+3)
BUT.....
If you multiply them, you get twice the original function. You have to divide by 2.

Answer 3

f(x) = 2x^3 + 6x^2 - 4.5x - 13.5

f(-3) = 2(-3)^3 + 6(-3)^2 - 4.5(-3) - 13.5 = -54 +54 +13.5 - 13.5 = 0

so x = -3 is one zero

divide f(x) by x + 3

x+3)2x^3+6x^2-4.5x-13.5(2x^2
......2x^3+6x^2
_____________________
......................-4.5x-13.5(-4.5
.......................-4.5x-13.5
______________________
...............................0

so the quotient is 2x^2 - 4.5

to find out the zeros equate it zero

2x^2 - 4.5 = 0

2x^2 = 4.5

x^2 = 2.25

x = + sqrt(2.25) or - sqrt(2.25)

x = +1.5 or -1.5


So all the 3 zeros of f(x) are 1.5, - 1.5 and -3