Download Section 2.3 Real Zeros of Polynomial Functions Important Vocabulary

Section 2.3
27
Real Zeros of Polynomial Functions
Section 2.3 Real Zeros of Polynomial Functions
Objective: In this lesson you learned how to use long division and
synthetic division to divide polynomials by other
polynomials and how to find the rational and real zeros of
polynomial functions.
Important Vocabulary
Course Number
Instructor
Date
Define each term or concept.
Long division of polynomials A procedure for dividing two polynomials, which is
similar to long division in arithmetic.
Division Algorithm If f(x) and d(x) are polynomials such that d(x) ≠ 0, and the degree
of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x)
and r(x) such that f(x) = d(x)q(x) + r(x) where r(x) = 0 or the degree of r(x) is less than
the degree of d(x).
Synthetic division A shortcut for long division of polynomials when dividing by
divisors of the form x – k.
Remainder Theorem If a polynomial f(x) is divided by x – k, then the remainder is
r = f(k).
Factor Theorem A polynomial f(x) has a factor (x – k) if and only if f(k) = 0.
Upper bound A real number b is an upper bound for the real zeros of f if no real
zeros of f are greater than b.
Lower bound A real number b is a lower bound for the real zeros of f if no real zeros
of f are less than b.
I. Long Division of Polynomials (Pages 116−118)
When dividing a polynomial f(x) by another polynomial d(x), if
the remainder r(x) = 0, d(x)
divides evenly
The rational expression f(x)/d(x) is improper if . . .
into f(x).
What you should learn
How to use long division
to divide polynomials by
other polynomials
the
degree of f(x) is greater than or equal to the degree of d(x).
The rational expression r(x)/d(x) is proper if . . .
the degree
of r(x) is less than the degree of d(x).
Before applying the Division Algorithm, you should . . .
write the dividend and divisor in descending powers of the
variable and insert placeholders with zero coefficients for
missing powers of the variable.
Example 1: Divide 3x 3 + 4 x − 2 by x 2 + 2 x + 1 .
3x − 6 + (13x + 4)/(x2 + 2x + 1)
Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
28
Chapter 2
II. Synthetic Division (Page 119)
Can synthetic division be used to divide a polynomial by x2 − 5?
Explain.
No, the divisor must be in the form x − k.
Polynomial and Rational Functions
What you should learn
How to use synthetic
division to divide
polynomials by binomials
of the form (x − k)
Can synthetic division be used to divide a polynomial by x + 4?
Explain.
Yes, rewrite x + 4 as x − (− 4).
Example 2: Fill in the following synthetic division array to
divide 2 x 4 + 5 x 2 − 3 by x − 5. Then carry out the
synthetic division and indicate which entry
represents the remainder.
5 2 0 5 0 −3
10 50 275 1375
2
10 55 275 1372 ← remainder
III. The Remainder and Factor Theorems (Pages 120−121)
To use the Remainder Theorem to evaluate a polynomial
function f(x) at x = k, . . .
use synthetic division to divide
What you should learn
How to use the
Remainder and Factor
Theorems
f(x) by x − k. The remainder will be f(k).
Example 3: Use the Remainder Theorem to evaluate the
function f ( x) = 2 x 4 + 5 x 2 − 3 at x = 5.
1372
To use the Factor Theorem to show that (x − k) is a factor of a
polynomial function f(x), . . .
use synthetic division on f(x)
with the factor (x − k). If the remainder is 0, then (x − k) is a
factor. Or, alternatively, evaluate f(x) at x = k. If the result is 0,
then (x − k) is a factor.
Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
Section 2.3
29
Real Zeros of Polynomial Functions
List three facts about the remainder r, obtained in the synthetic
division of f(x) by x − k:
1) The remainder r gives the value of f at x = k. That is, r = f(k).
2)
If r = 0, (x − k) is a factor of f(x).
3)
If r = 0, (k, 0) is an x-intercept of the graph of f.
IV. The Rational Zero Test (Pages 122−124)
Describe the purpose of the Rational Zero Test.
The Rational Zero Test relates the possible rational zeros of a
polynomial with integer coefficients to the leading coefficient
What you should learn
How to use the Rational
Zero Test to determine
possible rational zeros of
polynomial functions
and to the constant term of the polynomial.
State the Rational Zero Test.
If the polynomial f(x) = anxn + an – 1xn – 1 + . . . + a2x2 + a1x + a0
has integer coefficients, every rational zero of f has the form:
rational zero = p/q, where p and q have no common factors other
than 1, p is a factor of the constant term a0, and q is a factor of
the leading coefficient an.
To use the Rational Zero Test, . . .
first list all rational
numbers whose numerators are factors of the constant term and
whose denominators are factors of the leading coefficient. Then
use trial and error to determine which of these possible rational
zeros, if any, are actual zeros of the polynomial.
Example 4: List the possible rational zeros of the polynomial
function f ( x) = 3 x 5 + x 4 + 4 x 3 − 2 x 2 + 8 x − 5 .
± 1, ± 5, ± 1/3, ± 5/3
Some strategies that can be used to shorten the search for actual
zeros among a list of possible rational zeros include . . .
using a programmable calculator to speed up the calculations, using a
graphing utility to estimate the locations of zeros, using the
Intermediate Value Theorem (along with a table generated by a
graphing utility) to give approximations of zeros, or using the Factor
Theorem and synthetic division to test possible rational zeros, etc.
Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
30
Chapter 2
V. Other Tests for Zeros of Polynomials (Pages 124−126)
State the Upper and Lower Bound Rules.
Let f(x) be a polynomial with real coefficients and a positive
leading coefficient. Suppose f(x) is divided by x − c, using
synthetic division.
Polynomial and Rational Functions
What you should learn
How to use Descartes’s
Rule of Signs and the
Upper and Lower Bound
Rules to find zeros of
polynomials
1. If c > 0 and each number in the last row is either positive
or zero, c is an upper bound for the real zeros of f.
2. If c < 0 and the numbers in the last row are alternately
positive and negative (zero entries count as positive or
negative), c is a lower bound for the real zeros of f.
Explain how the Upper and Lower Bound Rules can be useful in
the search for the real zeros of a polynomial function.
Explanations will vary. For instance, suppose you are checking a
list of possible rational zeros. When checking the possible
rational zero 2 with synthetic division, each number in the last
row is positive or zero. Then you need not check any of the other
possible rational zeros that are greater than 2 and can concentrate
on checking only values less than 2.
y
Additional notes
x
y
x
Homework Assignment
Page(s)
Exercises
Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.