Download Episode 3 Slides - Department of Mathematical Sciences

MATH 57091 - Algebra for High School Teachers
Factoring Polynomials with Rational Coefficients I: Tools
Professor Donald L. White
Department of Mathematical Sciences
Kent State University
D.L. White (Kent State University)
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Review
Given a polynomial f (x) ∈ Q[x], we would like to factor f (x),
as completely as possible, over Q, R, or C.
Recall some of the tools we introduced previously:
If f (x) ∈ Q[x] and d is a common multiple of the denominators
of the coefficients of f (x), then p(x) = d · f (x) is in Z[x].
Thus if p(x) = a(x)b(x), then f (x) = d1 a(x)b(x).
It therefore suffices to consider factoring polynomials
with integer coefficients over Q.
Gauss’s Lemma:
If f (x) ∈ Z[x] and f (x) = a1 (x)b1 (x), with a1 (x), b1 (x) ∈ Q[x],
then f (x) = a(x)b(x), with a(x), b(x) ∈ Z[x]
and a1 (x) = αa(x), b1 (x) = βb(x) for some α, β ∈ Q.
In order to factor polynomials with integer coefficients over Q,
it therefore suffices to consider factoring polynomials
with integer coefficients over Z.
D.L. White (Kent State University)
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Review
[Previously introduced tools, continued]
Factor Theorem:
Let f (x) ∈ F [x], where F is a field.
An element α of F is a root of f (x) if and only if
x − α is a factor of f (x).
Let F be a field and f (x) ∈ F [x] with deg f (x) = n.
If the leading coefficient of f (x) is c
and f (x) has n roots r1 , r2 , . . . , rn in F ,
then
f (x) = c(x − r1 )(x − r2 ) · · · (x − rn ).
We will now concentrate on finding roots (in Z, Q, R, or C)
of polynomials in Z[x].
D.L. White (Kent State University)
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Tools for Finding Roots
Our first tool for finding roots of polynomials is the Quadratic Formula.
Quadratic Formula:
If f (x) = ax 2 + bx + c ∈ C[x] with a 6= 0, then the roots of f (x) are
√
√
−b − b 2 − 4ac
−b + b 2 − 4ac
and
.
2a
2a
The number D = b 2 − 4ac is the discriminant of f (x).
The roots of f (x) are always in C.
If b 2 − 4ac 6= 0, then there are two distinct roots.
If b 2 − 4ac = 0, then there is one root of multiplicity 2.
Let f (x) ∈ Z[x].
The roots of f (x) are in R ⇐⇒ b 2 − 4ac > 0.
The roots of f (x) are in Q ⇐⇒ b 2 − 4ac is a perfect square.
There are also (much more complicated) formulas for the roots
of cubic and quartic polynomials. (See §3.6 of the text.)
D.L. White (Kent State University)
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Tools for Finding Roots
The graph of a polynomial can also be useful in locating roots.
Let f (x) ∈ R[x]. The graph of f (x) is the set of points (a, b)
in the plane such that f (a) = b.
For a real number α, the following are equivalent:
i
ii
iii
x = α is a root of f (x) in R.
x − α is a factor of f (x) in R[x].
The graph of f (x) intersects the x-axis at x = α.
Moreover,
if x = α is a root of odd multiplicity,
then the graph of f (x) crosses the x-axis at x = α, and
if x = α is a root of even multiplicity,
then the graph of f (x) touches the x-axis at x = α.
D.L. White (Kent State University)
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Rational Root Test
Theorem [Rational Root Test]
Let
f (x) = an x n + an−1 x n−1 + · · · + a1 x + a0
be a polynomial with integer coefficients.
If
r
s
is a rational root of f (x), with r , s ∈ Z and (r , s) = 1, then
r | a0 and s | an .
If an = 1, then s = ±1 in the theorem, and we have the following corollary.
Corollary
If
f (x) = x n + an−1 x n−1 + · · · + a1 x + a0
is a monic polynomial with integer coefficients,
then any rational root of f (x) must be an integer r such that r | a0 .
D.L. White (Kent State University)
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Rational Root Test
Proof of Theorem: Let sr , with (r , s) = 1, be a rational root of f (x).
We then have
r n−1
r r r n
+ an−1
+ · · · + a1
f
= an
+ a0
s
s
s
s
r n−1
r
rn
= an · n + an−1 · n−1 + · · · + a1 · + a0 = 0.
s
s
s
Multiplying both sides of the last equation by s n , we obtain
an r n + an−1 r n−1 s + · · · + a1 rs n−1 + a0 s n = 0.
Hence
an r n = − an−1 r n−1 s + · · · + a1 rs n−1 + a0 s n ,
so s | an r n . But (s, r ) = 1, so (s, r n ) = 1, and s | an by Euclid’s Lemma.
Similarly,
a0 s n = − an r n + an−1 r n−1 s + · · · + a1 rs n−1 ,
hence r | a0 s n and (r , s n ) = 1, which implies r | a0 .
D.L. White (Kent State University)
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Example
EXAMPLE: Find all possible rational roots of f (x) = 2x 8 + 3x 3 + 6.
Solution: The possible rational roots are the numbers ± sr , where
r | 6, s | 2, (r , s) = 1,
and r , s are positive.
Hence
r
= 1, 2, 3, 6;
s = 1, 2,
and the possible rational roots are
±1, ±2, ±3, ±6, ± 12 , ± 23 .
(Of course, at most 8 of these can be roots,
and possibly no “possible” root is actually a root.)
D.L. White (Kent State University)
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