<PRE><B>Question 628062</B>
&lt;pre&gt;
f(x) = x&lt;sup&gt;4&lt;/sup&gt; + 3x&lt;sup&gt;2&lt;/sup&gt; + 5

Synthetic division

To do synthetic division with this polynomial you must insert
zero placeholders for the missing 3rd and 1st powers of x.
of x:

f(x) = x&lt;sup&gt;4&lt;/sup&gt; + 0x&lt;sup&gt;3&lt;/sup&gt; + 3x&lt;sup&gt;2&lt;/sup&gt; + 0x + 5

-3|1  0  3   0   5
  |&lt;u&gt;  -3  9 -36 108&lt;/u&gt;
   1 -3 12 -36 &lt;font color=&quot;red&quot;&gt;113&lt;/font&gt;

Answer: f(-3) = &lt;font color=&quot;red&quot;&gt;113&lt;/font&gt;

Check by direct substitution:

f(x) = x&lt;sup&gt;4&lt;/sup&gt; + 3x&lt;sup&gt;2&lt;/sup&gt; + 5

f(-3) = (-3)&lt;sup&gt;4&lt;/sup&gt; + 3(-3)&lt;sup&gt;2&lt;/sup&gt; + 5

f(-3) = 81 + 3·9 + 5 = 81 + 27 + 5 = &lt;font color=&quot;red&quot;&gt;113&lt;/font&gt;

So it checks.  The remainder when using a number with synthetic division
will always give the same answer as substituting that number for x directly.

Edwin&lt;/pre&gt;

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