46 2. THE LIBRARY OF FUNCTIONS
Knowledge Box 2.8
Suppose that f .x/ is a polynomial and that f .c/ D 0 for some number
c. en .x c/ is a factor of f .x/.
is result is called the root-factor theorem for polynomials. If we’re trying to factor a
polynomial, one approach is to plug in numbers looking for a root (graphing the polynomial
can narrow down the possibilities). Another way to state the root-factor theorem is the following.
Knowledge Box 2.9
Suppose that f .x/ is a polynomial and that f .c/ D 0 for some number
c. en for some polynomial g.x/ we have that
f .x/ D .x c/g.x/:
e second book in this series, Fast Start Integral Calculus contains a chapter that goes into far
more detail about the properties of polynomials—something we can do once we have the tools
of calculus at our fingertips.
PROBLEMS
Problem 2.5 For each of the following functions, determine if it is a polynomial. You may
need to simplify the function to tell if it is a polynomial.
1. f .x/ D 1 C .x C 1/ C .x
2
C x C 1/C
.x
3
C x
2
C x C 1/
2. g.x/ D .x
2
C 1/
3
C
.x
2
x C 1/
2
C 7
3. h.x/ D x C 7
4. r.x/ D 17
5. s.x/ D
x
x
2
C 1
6. q.x/ D x
3
C 4:1x
2
3:2x
2
C
4:6x 3:8
7. a.x/ D
x
3
C x C 1
x
2
C 1
1
x
2
C 1
8. b.x/ D .x
2
C
p
x/.x
2
p
x/
9. c.x/ D .x
2
C
p
x/
2
10. d.x/ D
1
x
2
x
2
C
3x
3
x C 2
x
2
2.1. POLYNOMIALS 47
Problem 2.6 Place each of the following polynomials into standard form.
1. f .x/ D 1 C .x C 1/ C .x C 1/
2
2. g.x/ D .x
2
C x C 1/
2
3. h.x/ D .x
2
1/.x
2
C 1/.x C 2/
2
4. r.x/ D x.x C 1/.x C 2/.x C 4/
5. s.x/ D .x
2
C 1/.x C 1/C
.x
2
C 1/.x 2/ C .x
2
C 1/
2
6. q.x/ D x
3
.x C 1/
3
.x 1/
3
7. a.x/ D .x C 1/
3
.x 1/
3
8. b.x/ D .x 2/
3
.x
3
6x
2
C 12x/
Problem 2.7 Find the degree of each of the polynomials in Problem 2.6.
Problem 2.8 Give a simple rule for telling if a polynomial is an odd function, an even function,
or neither.
Problem 2.9 Find the range of each of the following polynomials.
1. y D 3
2. y D 3x C 1
3. y D x
2
C 6x C 12
4. y D x
3
C 3x
2
7x C 8
5. y D x
4
C 5x
2
6. y D .x C 1/.x C 2/.x C 3/.x
2
C 1/
7. y D .x
2
C 5/
3
Problem 2.10 Argue convincingly that a positive whole number power of a polynomial is a
polynomial.
Problem 2.11 Find a polynomial of degree 3 with one root at x D 1.
Problem 2.12 Find a polynomial of degree 4 with no roots at x D 1.
Problem 2.13 Find a polynomial of degree 3 with roots at x D 0; ˙3.
Problem 2.14 Find a polynomial of degree 3 with roots at x D 0; ˙2 so that f .3/ D 5.
48 2. THE LIBRARY OF FUNCTIONS
Problem 2.15 Prove that if two polynomials both have a root at x D a , then so does their sum.
Problem 2.16 Suppose that we have several polynomials. Fill in the box in the following sen-
tence. e set of roots of the product of the polynomials is the of the sets of roots of
each of the polynomials.
Problem 2.17 True or false (and explain): if we divide two polynomials the resulting function
is a polynomial.
Problem 2.18 Demonstrate that the result of dividing two polynomials can be a polynomial.
Problem 2.19 Given that each of the following graphs is the graph of a polynomial, give as
much information about the degree, coefficients, and number and value of roots as you can.
1.
5.0
-5.0
5.0-5.0
2.
5.0
-5.0
5.0-5.0
3.
5.0
-5.0
5.0-5.0
4.
5.0
-5.0
5.0-5.0
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