Page 1 RICE UNIVERSITY Fixed-Polynomial Approximate Spectral Transformations for
Using Technology to Approximate Roots of Polynomial Equations.
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Transcript of Using Technology to Approximate Roots of Polynomial Equations.
Using Technology to Approximate Roots Using Technology to Approximate Roots of Polynomial Equationsof Polynomial Equations
3 2
You already know how to solve linear and quadratic equations.
In this section we will discuss how to solve a polynomial equation
of higher degree, such as 5 2 0 by using
a graphing calculator.
x x x
3 2Example 1. Solve 5 2 0 by using a graphing calculator.x x x
Let f denote a continuous function. If a<bAnd if f(a) and f(b) are of opposite sign, then the graph of f has at least one zero between a and b.
Intermediate Value Theorem
To get a more precise answer, you can change the table setting.
Solving Polynomial Equations by FactoringSolving Polynomial Equations by FactoringFactoring by grouping
Ex. 1. Solve:3 25 4 20 0x x x
3 2( 5 ) (4 20) 0x x x
2 5 4 5 0x x x 2 4 5 0x x
2 2 5 0x x x
2, 2, 5x x x
Variable substitutionAnother method of solving higher-degree polynomial equations involves recognizing polynomials that have quadratic form.
Ex. 2. Solve 4 22 3 0x x 2Let y x 22 22 3 0x x
22 3 0y y
2 3 1 0y y 3
2y 2 3
2x
3 6
22x
or 1y 2 1x
x i
The Rational Root Theorem
Let P(x) be a polynomial of degree n with integral coefficients and a nonzero constant term:
0
11 0, 0...n n
n n where aP x a x a x a
If one of the roots of the equation P(x) = 0 is where
P and q are nonzero integers with no common factor other than 1, then p must be a factor of and q must be a factor of
px
q
0ana
Ex. 3.
a. According to the rational root theorem, what are the possible rational roots of 4 3 23 13 15 4 0P x x x x
px
q is a possible rational root if p is a factor of –4 and q is a
factor of 3.
1, 2, 4
1, 3
p
q
1 2 41, 2, 4, , ,
3 3 3or
b. Determine whether any of the possible rational roots really are roots. Then find all other roots, real or imaginary.
Try synthetic substitution
Or substitute in to see if any P(x) = 0
4 3 21 3 1 13 1 15 1 4 27P
4 3 21 3 1 13 1 15 1 4 1P
4 3 22 3 2 13 2 15 2 4 208P
4 3 22 3 2 13 2 15 2 4 0P
Use synthetic division 2 3 13 15 0 4
3
6
7
14
1
22
4
0
3 22 3 7 2x x x x
Find the other roots of 3 23 7 2P x x x x
Check x = -2 again because it may be a double root
2 3 7 1 2
36
1
2
1
2
0
1 1 4 3 1 1 13
6 6x
23 1 0x x
1 132( ),
6So x a double root and x