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Have you ever wondered if there's a quick way to find the remainder of a polynomial division without performing long division or the lengthy Horner's method? The answer lies in the Remainder Theorem!
The Remainder Theorem provides an interesting relationship between the remainder of a polynomial division and the value of the polynomial itself.
If a polynomial is divided by , then the remainder is .
This means that to find the remainder when is divided by , we simply need to evaluate the polynomial at .
This theorem stems directly from the polynomial division algorithm we already know:
Where:
Now, let's substitute into the division algorithm equation:
This proves that the value of the polynomial at is equal to the remainder when is divided by .
The Remainder Theorem is very useful for quickly determining the remainder of a division or for evaluating a polynomial at a specific point.
Find the remainder if is divided by .
The divisor is , or , so .
Coefficients of (completing the x term): .
From Horner's method, we get:
According to the Remainder Theorem, the remainder when is divided by is .
Let's calculate :
The result is the same! Using the Remainder Theorem, we found the remainder is , just like with Horner's method, but without performing the full division process.
This shows that evaluating is another way to find the remainder of division by .
If is divided by , determine the remainder using the Remainder Theorem.
According to the Remainder Theorem, the remainder when is divided by is .
So, the remainder is .
