96 3. THE ARITHMETIC, GEOMETRY, AND CALCULUS OF POLYNOMIALS
x
5
D0.32237954
x
6
D0.32221744
x
7
D0.32219065
x
8
D0.32218623
x
9
D0.32218549
x
10
D0.32218537
x
11
D0.32218535
x
12
D0.32218535
At which point the number has stopped changing and so is the root to 8 decimals. You can verify
that f .x/ has only one root by graphing it.
˙
e picture on the cover of this book demonstrates how Newton’s method can be used to make
pretty pictures. It plots points on the complex plane with the real part of a number plotted on
the x-axis, and the imaginary part on the y-axis. It uses the polynomial:
x
6
C 2:43 x
4
5:8644 x
2
10:4976
is polynomial has six roots. Points are colored based on which root they converge to when
used as a starting guess for Newton’s method.
PROBLEMS
Problem 3.6 Compute the following polynomial products.
1. .x
3
C 2x
2
C x C3/ .x C5/
2. .x
2
C 4x 1/ .x
2
4x C 1/
3. .x
3
C 6x
2
C 11x C6/ .x
3
6x
2
C 11x 6/
4. .x
3
C 3x C7/ .x
4
C x C2/
5. .x C 1/
3
.x C2/
3
6. .x
2
C x C1/
3
3.1. POLYNOMIAL ARITHMETIC 97
Problem 3.7 Perform the following polynomial divisions.
1. .x
4
C 4x
3
C 6x
2
C 4x C 1/ .x C 1/
2. .x
4
x
3
C 2x
2
C x C3/ .x
2
C x C1/
3. x
4
C x
3
C 2x
2
C x C1/ .x
2
C 1/
4. .x
5
C 8x
4
C 21x
3
C 35x
2
C 28x C15/ .x
2
C 2x C3/
5. .x
6
x
5
C 2x
4
x
3
C 2x
2
x C1/ .x
2
x C1/
6. .x
6
1/ .x 1/
Problem 3.8 Use Newton’s method to approximate the roots of the polynomial
f .x/ D x
3
3x C1
using initial guesses of x
0
D 2; 0; and 2. ese should generate three distinct roots.
Problem 3.9 Use Newton’s method to find a root of:
g.x/ D x
5
5x
2
6
Problem 3.10 Earlier in the text it was asserted that a polynomial of odd degree must have at
least one root. Explain why.
Problem 3.11 e polynomials
f .x/ D x
2
2x C1
and
g.x/ D xr 3x C 2
both have a root at x D 1. Use a spreadsheet to apply Newton’s method to both of these problems
with an initial guess of x
0
D 0:4. Both these intial guesses should converge to x D 1 – what is
different about the convergence to one for these two polynomials. Why? Pictures may help.
98 3. THE ARITHMETIC, GEOMETRY, AND CALCULUS OF POLYNOMIALS
Problem 3.12 Find all real roots of the following polynomials.
1. f .x/ D x
4
2x
3
2x
2
2x 3
2. g.x/ D x
3
6x
2
C 11x 6
3. h.x/ D x
6
64
4. q.x/ D x
4
5x
2
C 4
5. r.x/ D x
4
8x
2
C 5x C6
6. s.x/ D x
3
6x
2
C 12x 8
Problem 3.13 Find a polynomial of the form
x
4
C ax
3
C bx
2
C xc C d
with none of a; b; c; or d zero that has no real roots at all.
Problem 3.14 Suppose that p.x/ is a polynomial. How many roots does p.x/
2
C 1 have?
Justify your answer with one or more sentences.
Problem 3.15 Consider the polynomial
q.x/ D x
3
25x
First find its three roots. en figure out how many roots q.x/
2
1 has.
Problem 3.16 Find a polynomial with roots at x D 1; 2; 3; 4; and 5.
Problem 3.17 Find a polynomial with roots at x D 1; 2; 3; 4; and 5 that does not take on any
negative values.
Problem 3.18 Given the Newton’s method formula for a polynomial:
x
iC1
D
2x
3
i
x
2
i
C 4
3x
2
i
2x
i
4
what is the polynomial?
Problem 3.19 If the Newton’s method formula for a polynomial is
x
iC1
D
3x
4
i
C x
2
i
7
4x
4
i
C 10x
i
what is the polynomial?
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