Mathematical Investigations III
Name:
Mathematical Investigations III - A View of the World
Long Division Practice
You learned how to do long division with integers while in grade school and probably learned how
to do polynomial long division in Algebra I. Here is an example of polynomial long division.
4
3
2
x  6x  4x  11x  2
x3
x 3  3x 2  5x  4 

10
x3
x  3 x 4  6x 3  4x 2  11x  2
Long Division:

 x 4  3x 3
(1) Find the monomial (single-term polynomial)
you must multiply the leading term of the
divisor to get the leading term of the dividend
and write that as part of your answer.
(2) Subtract from the dividend the product of the
monomial you just wrote and the divisor to get
the new dividend.
(3) Repeat Steps 1 and 2 until Step 1 is impossible,
that is, when the degree of the divisor is greater
than the degree of the dividend.

 3x  4x 2
3

 3x 3  9x 2

 5x  11x
2

 5x 2  15x

 4x  2

 4x  12

 10
Quotient: x 3  3x 2  5 x  4
Remainder:  10
The remainder of the division is the final dividend when the procedure stopped. The quotient is
the sum of all the monomials you found executing Step 1. The goal of long division is to be able to
DIVIDEND
REMAINDER
write
as QUOTIENT 
.
DIVISOR
DIVISOR
1. Find the quotient and the remainder for each division problem using long division.
2x 3  5x 2  3x  6
(a)
x2
3x 4  10x 3  30x  3
(b)
x4
Poly 11.1
Rev. F10
Mathematical Investigations III
Name:
x  6x  2x  3x  1
x 2  2x  3
4
(c)
3
2
(d)
6x 3  15x 2  87x  30
2x  5
The division problems we have done were possible due to the following theorem from algebra.
The Division Algorithm
Assume all polynomials have complex number coefficients. For any polynomial P(x) and
divisor polynomial D(x), there exists a quotient polynomial Q(x) and remainder polynomial
P(x)
R(x)
R(x) such that
where the degree of R (x ) is less than the degree of D(x) .
 Q(x) 
D(x)
D(x)
This is equivalent to P( x)  D( x)  Q( x)  R( x) with the same condition on the degree of R (x ) .
Also, P(x) factors as D( x)  Q( x) , and hence is divisible by D(x) , if and only if R(x) = 0.
2. Rewrite the last two parts of the previous question with quotients and remainders in the form
P( x)  D( x)  Q( x)  R( x)
4
3
2
(a) x  6x  2x  3x  1 =
3. (+) Find all integer values of n for which
3
2
(b) 6x  15x  87x  30 =
n 2  3n  7
is an integer.
n4
Poly 11.2
Rev. F10