This section gives a useful application of synthetic division
and the division algorithm,
called the Remainder Theorem or the Polynomial Remainder Theorem.
The Remainder Theorem provides an efficient way to evaluate a polynomial at a given number:
that is, to find $\,P(c)\,$ for a given polynomial $\,P(x)\,$ and a given real number $\,c\,$.
How?
Just divide $\,P(x)\,$ by $\,x - c\,$ and then take the remainder!
After the example below is a precise statement of the Remainder Theorem, and why it works.
Let $\,P(x) = 2x^5 - 3x^4 + 5x - 7\,$. Find $\,P(4)\,$.
SOLUTION:
Divide $\,P(x)\,$ by $\,x-4\,$, using synthetic division.
Since we're dividing by $\,x - 4\,$, the number $\,4\,$ goes in the little box:
Let's check this result with old-fashioned function evaluation:
$\displaystyle
\begin{align}
\cssId{s17}{P(4)\ }
&\cssId{s18}{= 2\cdot4^5 - 3\cdot4^4 + 5\cdot 4 - 7}\cr
&\cssId{s19}{= 2\cdot1024 - 3\cdot256 + 20 - 7}\cr
&\cssId{s20}{= 1293}
\end{align}
$
Notice that function evaluation requires powers (like $\,4^5\,$),
whereas the Remainder Theorem requires only multiplication and addition.
Let $\,P(x)\,$ be a polynomial, and let $\,c\,$ be a real number. Then, $\,P(c)\,$ is the remainder when $\,P(x)\,$ is divided by $\,x - c\,$. |
PROOF:
By the division algorithm,
there exist unique polynomials $\,Q(x)\,$ (the quotient) and $\,R(x)\,$ (the remainder)
such that
$$\cssId{s30}{P(x) = Q(x)(x-c) + R(x)}\,,$$
where either