Math 135
Polynomial Division
Examples
Before we begin working with the graphs of polynomial and rational functions we need
to develop the tools with which we will extract the information necessary to draw our
graphs. We will be using the following definition:
Definition 1. (Polynomial function) A polynomial function is a function of the form
p(x) = an xn + an−1 xn−1 + an−2 xn−2 + · · · + a0 x0
and satisfies these conditions
• The numbers an , an−1 , . . . , a0 , called the coefficients of p(x), are real numbers,
• Each exponent of x is a nonnegative integer.
We call the term with the highest power of x the leading term of p(x) and the coefficient
of the leading term is called the leading coefficient of p(x). The degree of the leading term
is called the degree of the polynomial.
Example 2. p(x) = 5x2 −x+10x10 −1 is a polynomial function with leading term 10x10 . The
leading coefficient if 10 and the degree of p(x) is 10. It is understood that the coefficients
of the powers of x not present are zero, i.e. writing p(x) in the standard form given in the
definition
p(x) = 5x2 − x + 10x10 − 1 = 10x10 + 0x9 + 0x8 + 0x7 + 0x6 + 0x5 + 0x4 + 0x3 + 5x2 − 1x1 − 1x0
Definition 3. (Rational Function) A function which is the quotient of two polynomial functions is called a rational function. That is, both the numerator and the denominator must
be polynomial functions.
3
g(x)
+x+1
= 3x x−1
is a rational function because both g(x) = 3x3 + x + 1
Example 4. f (x) = h(x)
and h(x) = x − 1 are polynomials.
A natural question is to ask whether the above f (x) has also a leading term. The
answer requires to express f (x) as a polynomial and hence we must be able to divide one
polynomial by another.
g(x)
The DIVISION LAW states that if f (x) = h(x)
is a rational function and the degree of
g(x) is at least as large as the degree of h(x), then there are unique polynomials q(x) and
r(x) such that
r(x)
g(x)
f (x) =
= q(x) +
h(x)
h(x)
Moreover, the degree of r(x) is strictly less than the degree of h(x). We call the polynomial
h(x) the divisor, the polynomial q(x) the quotient, and the polynomial r(x) the remainder.
University of Hawai‘i at Mānoa
120
R Spring - 2014
Math 135
Polynomial Division
Examples
Let us apply the division law to the above example. We dive 3x2 + x + 1 by x − 1 using
polynomial long division:
3x + 4
x − 1 )3x2 + x + 1
3x2 − 3x
4x + 1
4x − 4
5
The quotient 3x + 4 and the divisor x − 1 are degree one polynomials and the remainder
5 is a degree zero polynomial and
5
3x2 + x + 1
= 3x + 4 +
x−1
x−1
The leading term of f (x) is the leading term of the quotient and it is just the quotient of
the leading terms of g(x) and h(x). If the remainder is zero, then we say that h(x) divides
g(x) evenly. In this case h(x) is a factor of f (x).
The FACTOR THEOREM states that a is a root of a polynomial p(x) if and only if
(x − a) is a factor of p(x). That is, if and only if (x − a) divides p(x) evenly. Moreover, p(c)
is the remainder of the division of p(x) by the linear polynomial (x − c).
Example 5. Factor t(x) = x3 − x2 − 2x + 2 into linear factors provided that 1 is one root.
Since 1 is a root, it follows by the factor theorem that x − 1 divides t(x) evenly and
t(1) = 0. By polynomial long division we obtain
x3 − x2 − 2x + 2
= x2 − 2 ⇒ x3 − x2 − 2x + 2 = (x2 − 2)(x − 1)
x−1
and since we have a difference of squares t(x) factors further as
√
√
(x − 2)(x + 2)(x − 1)
√
√
Observe that the roots of t(x) are {− 2, 1, 2}.
University of Hawai‘i at Mānoa
121
R Spring - 2014
Math 135
Polynomial Division
Worksheet
1. Write the following polynomials in standard form:
(a) p(x) = x3 − x + 1
(b) g(x) = x6 + x2 − 11
(c) f (x) = 5
2. Perform the division. List the divisor, quotient, and remainder.
(a)
x6 −64
x−2
(b)
4x3 +x+1
2x2 +x+1
(c)
4x2 +3x+7
x2 −2x+1
3. Factor and find all roots:
(a) x3 + 8x2 − 3x − 24, if −8 is a root.
(b) 2x3 + x2 − 5x − 3, if − 32 is a root.
(c) x3 − 7x2 − 4x + 28, if 7 is a root.
Sample Midterm
University of Hawai‘i at Mānoa
Sample Final
6 A B C D
22 A B C D
122
R Spring - 2014