Thread: help with polynomial roots question

1. help with polynomial roots question

Use synthetic division to show that all of the real zeros of f(x)=x^3-3x^2-4x+5 lie in the interval [-2,4]

2. Hello, cityismine!

Use synthetic division to show that all of the real zeros of $\displaystyle f(x) \:=\: x^3-3x^2-4x+5$
lie in the interval $\displaystyle [-2,4]$
I'm not sure why Synthetic Divison is used here . . .

We find that: .$\displaystyle f(-2) \:=\:-8-12+8+5 \:=\:-7$
. . The graph is below the x-axis.

We find that: .$\displaystyle f(0) \:=\: 0 - 0 - 9 + 5 \:=\:5$
. . The graph is above the x-axis.

We find that: .$\displaystyle f(2) \:=\:8 - 12 - 8 + 5 \:=\:-7$
. . The graph is below the x-axis.

We find that:. $\displaystyle f(4) \:=\:64 - 48 - 16 + 5 \:=\:5$
. . The graph is above the x-axis.

So far, the graph looks like this:
Code:
            |
|         (4,5)
(0,5)*           *
|           :
-2     |     2     :
--+-----+-----+-----+----
:     |     :     4
:     |     :
:     |     :
*     |     *
(-2,-7)  |   (2,-7}
|

Since a polynomial function is continuous, the graph must cross the x-axis
. . on the intervals $\displaystyle (-2,0),\:(0,2).\:(2,4)$
And those are the three (real) zeros of the cubic function.
Code:
            |
*           *
* | *         :
-2     |     2     :
--+--o--+--o--+---o-+----
:     |     :     4
:     |     :  *
: *   |   * : *
*     |     *
|