On the Use of the Remainder Theorem
The Remainder Theorem is covered in virtually every College Algebra and/or Precalculus course.
In our text it has an immediate use, namely
to produce the Factor Theorem. However, the Remainder Theorem does strike most students as being a
sort of silly theorem. After all, a student's experience is that in doing a division, the "big"
answer is the quotient, and the remainder merely shows the difference between an "exact" division and
the particular problem at hand. Furthermore, with the use of a powerful
calculator, if we want to know the remainder, we just ask the calculator to do the division.
It will produce both the quotient and the remainder. The
POLYDIV and POLYN programs for the TI-83 do this,
the POLY3 program for the TI-85 and TI-86 does this, and
the TI-89 and TI-92 do such divisions as part of the standard functionality of the calculator.
Based on experience and use, the Remainder Theorem would seem to be almost unimportant.
Such a feeling is not far from the truth. However, we do need to understand the
tremendous importance of the Remainder Theorem to generations of mathematicians and
math students.
If we do not have a powerful calculator, perhaps not even a simple calculator, then the Remainder Theorem
becomes an amazing tool in working with higher degree polynomials.
If we have a polynomial function that we want to factor, such as
x5 – 5x4 – 15x3 +
85x2 – 26x – 120
and we might even want to try such factors as (x+1), (x+2),
(x+3), (x+4), (x+5), (x+6), (x+8), (x+10),
(x–1), (x–2), (x–3), (x–4), (x–5), (x–6),
(x–8), (x–10), and others
we do not want to go through the long division algorithm for
each possible factor to see if the possible factor is
indeed a factor. We would much rather calculate the value of
f(– 1),
f(– 2),
f(– 3),
f(– 4),
f(– 5),
f(– 6),
f(– 8),
f(– 10),
f(1),
f(2),
f(3),
f(4),
f(5),
f(6),
f(8),
f(10), and so on
looking for the case or cases where the value of the polynomial function is 0, because, according to the
Remainder Theorem, that means that the remainder of the corresponding division would be zero, and that means,
according to the Factor Theorem, that the corresponding binomial is indeed a factor of the original polynomial.
Thus, for example, it is the case that 35 – 5(34) – 15(33) +
85(32) – 26(3) – 120 = 0
Therefore, we know that (x–3) is a factor of f(x).
Of course, we do not know the quotient polynomial, but we do know
that if we do the long division then it will come out even, with no remainder.
Remembering that we are talking about doing this by hand, we will spend the effort to
do the long division to obtain that quotient,
knowing that our effort will be rewarded.
PRECALCULUS: College Algebra and Trigonometry
© 2000 Dennis Bila, James Egan, Roger Palay