Question 230380
{{{ P(x)= x^3+7x^2+4x}}} for P(a) we will divide P(x) by x-a using synthetic division. For a = 2:
<pre>
2 |   1   7   4   0
---       2  18  44       
     ---------------
      1   9  22  44
</pre>
Since our remainder is 44, P(2) = 44.<br>
To understand why this works, let's think about division in general. 
Let P(x) = any polynomial function
Let (x-a) = the divisor
Let Q(x) = the quotient you get from dividing P(x) by (x-a)
Let R/(x-a) = the remainder you get from dividing P(x) by (x-a). (Note: With a divisor like (x-a) R will just be some number. There will be no x's in the remainder.)
In other words:
{{{P(x)/(x-a) = Q(x) + R/(x-a)}}}
Let's multiply both sides by (x-a):
{{{(x-a)(P(x)/(x-a)) = (x-a)(Q(x) + R/(x-a))}}}
On the right side we need to use the Distributive Property:
{{{(x-a)(P(x)/(x-a)) = (x-a)*Q(x) + (x-a)(R/(x-a))}}}
Now we can cancel out some (x-a)'s:
{{{cross((x-a))(P(x)/cross((x-a))) = (x-a)*Q(x) + cross((x-a))(R/cross((x-a)))}}}
leaving:
{{{P(x) = (x-a)*Q(x) + R}}}
Therefore
{{{P(a) = (a-a)*Q(a) + R}}}
Since (a-a) is zero and zero times anything is zero and zero + R is R:
{{{P(a) = R}}}
This is why the remainder, after dividing a polynomial, P(x), by (x-a) is the value of P(a). And synthetic division is a quick, simple, compact way to do this type of division.