Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Roots of equations
CMPE220 - Discrete Computational Structures
May 24, 2010
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Names of polynomials
◮
Degree 0: Constant
◮
Degree 1: Linear
◮
Degree 2: Quadratic
◮
Degree 3: Cubic
◮
Degree 4: Quartic
◮
Degree 5: Quintic
◮
Degree 6: Sextic
◮
Degree 7 and on: # degree # with terms
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Constant Functions
Linear Functions
Quadratic Functions
Cubic Functions
Quartic Functions
Quintic Functions
Sextic Functions
Constant Function
p(x) = c
has root(s) if c = 0 only.
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Constant Functions
Linear Functions
Quadratic Functions
Cubic Functions
Quartic Functions
Quintic Functions
Sextic Functions
Linear Functions
p(x) = mx + n root:
y = −m/n.
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Constant Functions
Linear Functions
Quadratic Functions
Cubic Functions
Quartic Functions
Quintic Functions
Sextic Functions
Quadratic Functions
p(x) = ax 2 + bc + c
Solutions possible,
CMPE220 - Discrete Computational Structures
◮
By completing the square
◮
By shifting ax 2
◮
By By Lagrange
resolvents
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Constant Functions
Linear Functions
Quadratic Functions
Cubic Functions
Quartic Functions
Quintic Functions
Sextic Functions
Cubic Functions
p(x) = ax 3 + bx 2 + cx + d
Solution possible
CMPE220 - Discrete Computational Structures
◮
By cubic version of
discrimant.
◮
By Cardano’s method.
◮
By Lagrange resolvents.
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Constant Functions
Linear Functions
Quadratic Functions
Cubic Functions
Quartic Functions
Quintic Functions
Sextic Functions
Quartic Functions
p(x) = ax 4 +bx 3 +cx 2 +dx +e
CMPE220 - Discrete Computational Structures
◮
General Solution is
possible by Ferrari’s
method
◮
One can try to see it as
product of two quadratic.
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Constant Functions
Linear Functions
Quadratic Functions
Cubic Functions
Quartic Functions
Quintic Functions
Sextic Functions
Quintic Functions
p(x) =
ax 5 + bx 4 + cx 3 + dx 2 + ex + f
CMPE220 - Discrete Computational Structures
◮
Galois therom helps to see
whether
it is reducible by
factorization
◮
For some Bring-Jerard
Cubics like,
p(x) = x 5 + ax + b are
solvable by variable
change
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Constant Functions
Linear Functions
Quadratic Functions
Cubic Functions
Quartic Functions
Quintic Functions
Sextic Functions
Sextic Functions
p(x) = ax 6 + bx 5 + cx 4 +
dx 3 + ex 2 + fx + g
CMPE220 - Discrete Computational Structures
◮
Galois therom helps to see
whether it is reducible by
factorization
◮
For some in
hypergeometric form, may
be written as Kampé de
Feriét eqs.
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Factor = Root
Plan
Step 1
Step 2
Step 3
Step 4
Step 5
Guide
There are two common methods.
◮
Analytic solution
◮
Numeric solution
We will mostly be dealing with analytic solutions here.
We will use numeric solution to support analytic solutions.
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Factor = Root
Plan
Step 1
Step 2
Step 3
Step 4
Step 5
These are all the same
◮
Solving a polynomial equation p(x) = 0
◮
Finding roots of a polynomial equation p(x) = 0
◮
Finding zeroes of a polynomial function p(x)
◮
Factoring a polynomial function p(x)
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Factor = Root
Plan
Step 1
Step 2
Step 3
Step 4
Step 5
Step-by-step
1. Put it into standart form. p(x) = 0
2. Know how many roots to expect.
3. If this is linear, quadratic use know methods.you are done!
4. else, find a factor. You may guess or use numerical methods.
Factorize equation by (x − r ). If you get a lower degree
equation, then return to 3.
5. If you can’t find root or factor anymore, switch to numerical
methods.
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Factor = Root
Plan
Step 1
Step 2
Step 3
Step 4
Step 5
Step 1 - put into standart form
◮
tidy you equation.
◮
◮
divide all by a constant.
◮
◮
ex. x 4 − 6x 3 = x − 2 to x 4 − 6x 3 − x + 2 = 0
ex. 8x 2 + 16x + 8 = 0 to x 2 + 2x + 1
find common denominator
◮
ex. (1/3)x 3 + (3/4)x 2 (1/2)x + 5/6 to
(1/12)(4x 3 + 9x 2 6x + 10)
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Factor = Root
Plan
Step 1
Step 2
Step 3
Step 4
Step 5
Step 2 - Know how many roots
◮
Fundamental Theorem of Algebra says; non-constant
polynomial has at least one root.
◮
Factor Theorem says if r is root, then (x-r) is a factor of
polynomial.
◮
Descartes’ Rule of Signs helps you to identify signs of roots.
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Factor = Root
Plan
Step 1
Step 2
Step 3
Step 4
Step 5
Step 3 - Solve it with generic methods
If your polynomial is one of below, you have already found solution.
For Linear equations, If y = mx + n then x = −n/m
For Quadratic equations, Roots are x =
√
−b± b2 −4ac
2a
For Cubic equations, is solvable by Monic Formula.
For Quartic equations, is solvable by Ferrari’s Method.
For Quintic, Sextic and Septic equations, barely solvable.
Some of them is irreducible by factorizing. Galois Theory let us to
see if it is reducible.
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Factor = Root
Plan
Step 1
Step 2
Step 3
Step 4
Step 5
Step 4 - Try to guess a root
◮
you may guess it
◮
you may use numeric methods for finding a root.
◮
Factorize it by (x − r )
◮
If you get a degree lower polynomial, return to step 3.
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Factor = Root
Plan
Step 1
Step 2
Step 3
Step 4
Step 5
Step 5 - Numerical methods
◮
So if you are here, you need to use numerical methods.
◮
You may also use matlab, or GNU Octave to solve them in
symbolic form.
>>> syms a b c x;
>>> solve('a*x^2 + b*x + c')
ans =
-(b + (b^2 - 4*a*c)^(1/2))/(2*a)
-(b - (b^2 - 4*a*c)^(1/2))/(2*a)
>>>
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Roots of equations
◮
We have absolute analytic solutions up to 4th degree.
(Including 4th)
◮
We can test others if factorization possible.
◮
One common trick is to guess a root then apply factorization
to polynom to get a lower degree one.
◮
Higher degrees is solvable by programming. (Matlab, Octave,
LiveMath)
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
http://en.wikipedia.org/wiki/Degree_of_a_polynomial
http://en.wikipedia.org/wiki/Linear_function
http://en.wikipedia.org/wiki/Quadratic_function
http://en.wikipedia.org/wiki/Cubic_function
http://en.wikipedia.org/wiki/Quartic_function
http://en.wikipedia.org/wiki/Quintic_function
http://en.wikipedia.org/wiki/Sextic_function
http://oakroadsystems.com/math/polysol.htm
http://en.wikipedia.org/wiki/Galois_theory
http://www.mathworks.com/access/helpdesk/help/toolbox/symbolic/solve.html
CMPE220 - Discrete Computational Structures
Roots of equations
Introduction
Functions
A Solving Guide
Conclusion
References
Thank you
Thank you
Questions ?
Onur Özkol - [email protected]
CMPE220 - Discrete Computational Structures
Roots of equations