help with polynomial roots question

• May 25th 2008, 09:02 PM
cityismine
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]
• May 26th 2008, 08:28 AM
Soroban
Hello, cityismine!

Quote:

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

I'm not sure why Synthetic Divison is used here . . .

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

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

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

We find that:. $f(4) \:=\:64 - 48 - 16 + 5 \:=\:5$
. . 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 $(-2,0),\:(0,2).\:(2,4)$
And those are the three (real) zeros of the cubic function.
Code:

```            |             *          *           * | *        :     -2    |    2    :     --+--o--+--o--+---o-+----       :    |    :    4       :    |    :  *       : *  |  * : *       *    |    *             |```