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Lecture 6

MATH135 Lecture 6: Computing with polynomials

Department
Mathematics
Course Code
MATH135
Professor
Roxane Itier Itier
Lecture
6

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MATH 135, Winter 2015
Examples: Computations with polynomials
1 Multiplication and long division
Example 1. (Adding and multiplying polynomials)
(a) Consider the polynomials fpxq  2xp13iqand gpxq  ?3x3p45iqx p2iqin
Crxs. Then
fpxqgpxq  ?3x3p65iqxp14iq,
2fpxqgpxqp4x26iqp?3x3p45iqx2iq?3x35ix p45iq,
fpxqgpxqp2?3qx4p3?3iqx3p810iqx2p15 12iqx.
2x p13iq
q ?3x30x2 p45iqx p2iq
2?3x4 p?33?3iqx3
0x30x2
p810iqx2 p19 7iqx
q p42iqx p55iq
fpxqgpxq  2?3x4 p?33?3iqx3 p810iqx2 p15 9iqx p55iq
(b) Consider fpxq  x4x2and gpxq  x22 in Z3rxs. (We adopt the convention that when
working with polynomials in Zprxs, a congruence class ras P Zpis simply written as a. With
this understanding, we have fpxq  x4x2x42x2.) Check that
fpxqgpxq  x42x22x4x21,
fpxq2 px4x2q2 px4q22px4qpx2qpx2q2x8x6x4,
fpxqgpxq  x6x42x2x6x4x2.
x4x2
q x22
x6x4
q 2x42x2
fpxqgpxq  x6x42x2
l
1