Long division of polynomials A procedure for dividing two polynomials ,
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
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
divisors of the form x – k.
Remainder Theorem If a polynomial f(x) is divided by x – k, then the
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
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 160-162)
When dividing a polynomial f(x) by another polynomial d(x), if
the remainder r(x) = 0, d(x) divides evenly into f(x).
The rational ex pression f (x)/d(x) is improper if . . . 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).
The result of a division problem can be checked by . . . multiplying the quotient by the divisor and adding any remainder
to see if the original dividend is obtained.
III. The Remainder and Factor Theorems (Pages
To use the Remainder Theorem to evaluate a polynomial
function f(x) at x = k, . . . use synthetic division to divide
f(x) by x - k. The remainder will be f(k).
Example 3: Use the Remainder Theorem to evaluate the
function f (x) = 2x4 + 5x2 - 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.
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.
What you should learn
How to use the
Remainder Theorem and
the Factor Theorem
IV. The Rational Zero Test (Pages 166-168)
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
and to the constant term of the polynomial.
State the Rational Zero Test. If the polynomial
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, and p = a factor of the constant term a0, and q = 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 ± 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, or
using the Factor Theorem and synthetic division to test possible
rational zeros, etc.
What you should learn
How to use the Rational
Zero Test to determine
possible rational zeros of
V. Bounds for Real Zeros of Polynomial Functions
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. 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.
What you should learn
How to determine upper
and lower bounds for
zeros of polynomial