Problem: (Stanford Putnam Practice) Find the remainder when you divide by .
Solution: Since they’re both divisible by , we first divide that out, and just remember to multiply the remainder by at the end. Let be the first polynomial and be the second. we have
for some polynomials with . We want to find . Consider the two roots of . Plugging them into the equation, we obtain
Evaluating at those two values, we find that
But since has degree less than , the only possible is the constant polynomial . Then the remainder is . QED.
Comment: This is a super important technique when it comes to polynomial division. Using the roots of and the fact that , we can hypothetically always determine this way, without dividing. This usually comes up when has nice roots, so if it doesn’t, look for a better way.
Practice Problem: (Stanford Putnam Practice) How can the quadratic equation
have three roots ? [Reworded]
Leave a Reply
You must be logged in to post a comment.