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Section 3.7 – The Real Zeros of a Polynomial Function
In section 3.3, we have seen that if the polynomial function is given in factored form it is
very easy to find the zeros and he function can be graphed without the calculator by
using the concepts of multiplicity and end behavior.
In this section we are dealing with graphing polynomial functions which have not been
given in the factored form. Sometimes it is very easy to factor, but some others not so
easy. We’ll learn how to:
Determine the number of zeros of a polynomial function,
Determine the possible rational zeros,
Use the information to factor the polynomial with factors involving integral
coefficients
Division Algorithm for Polynomials
f ( x)  q( x)( x  c)  R
(1)
Example 1: Find the quotient and remainder if
f ( x)  2 x3  x 2  2 x  3 is divided by
x  2 . Then, write f(x) as indicated in equation (1)
a) Use long division
b) Use synthetic division
Remainder Theorem
In the equation f ( x)  q( x)( x  c)  R (from last page)
Notice that for
xc
f (c)  q(c)(c  c)  R
f (c)  R
Let
is
f be a polynomial function. If f ( x)
f (c) .
is divided by
xc
, then the remainder
Example 2: Use the remainder theorem to find the remainder if
f ( x)  2 x3  x 2  2 x  3 is divided by x  2
Example 3: Use the remainder theorem to find the remainder if
f ( x)  2 x3  x 2  2 x  3 is divided by x 1
Now do synthetic division to check your answer.
Factor Theorem
Let f be a polynomial function. Then
xc
is a factor of
f ( x)
if and only if
f (c) = 0
Example 4: Use the factor theorem to write the polynomial of example 3 in factored
form.
Example 5: Use the factor theorem to determine whether x – c is a factor of f. If it is,
write f in factored form. Do #2, page 257
Example 6: Use the factor theorem to determine whether x – c is a factor of f. If it is,
write f in factored form. Do #4, page 257
What if they don’t give us a value for c? Are we going to try some numbers at random by
finding f(#) to see if it is zero and proceed from there to find the factorization?
If we were given a list of numbers as possible zeros we could use the remainder
theorem to check whether they are zeros, if they are, we can use synthetic division to
find the “other” factor, and then factor the polynomial.
The Rational Zeros Theorem will help us find the list of possible zeros.
Theorem: Number of Real Zeros
A polynomial function of degree n, n ≥ 1, has at most n real zeros.
Rational Zeros Theorem
Let b be a polynomial function of degree 1 or higher of the form
f ( x)  an x n  an 1 x n 1  ...  a1 x  a0 , a0  0 where each coefficient is an integer. If
p
, in lowest terms, is a rational zero of f , then p must be a factor of a0 and q
q
must be a factor of an .
Theorem
Every polynomial function (with real coefficients) can be uniquely factored into a
product of linear factors and/or irreducible quadratic factors.
Corollary
A polynomial function (with real coefficients) of odd degree has at least one real
zero.
Example 7: Tell the maximum number of real zeros, and find the potential zeros of a
polynomial function. Use the real zeros to factor the function.
Do #30, page 257
Example 8: Tell the maximum number of real zeros, and find the potential zeros of a
polynomial function. Use the real zeros to factor the function.
Do #34, page 257
Example 9: Tell the maximum number of real zeros, and find the potential zeros of a
polynomial function. Use the real zeros to factor the function.
Do #36, page 257
Solving Polynomial Equations
Example 10: Solve the equation in # 56, page 258
Example 11: Solve the equation in # 60, page 258
Intermediate Value Theorem
Let f denote a continuous function. If a  b and if f ( a ) and f (b ) are of
opposite sign, then f has at least one zero between a and b .
Example 12: Use the intermediate value theorem to show that each function has a zero
in the given interval. Do # 64, page 258
Section 3.8 – Complex Zeros. Fundamental Theorem of Algebra
Example 13: Let’s refer to Example 8 from 3.7.
What if we want to factor the polynomial into a product of linear factors? Can we do that?
Example 14: Let’s refer to Example 10 from 3.7.
What if we want to find all solutions to the equation, real and complex?
Complex Zeros of Polynomials with Real Coefficients
Let f(x) be a polynomial whose coefficients are real numbers. If r = a + bi is a zero
of f(x), then the complex conjugate a – bi is also a zero of f.
Example 15: Find the remaining zeros of a polynomial function. Do # 2, page 263
Example 15: Find the remaining zeros of a polynomial function. Do # 10, page 264
Example 16: Form a polynomial with real coefficients having the given degree and
zeros. Do # 12, page 264.
Example 17: Use the given zero to find the remaining zeros of the function. Do # 18,
page 264.
Example 18: Find all the zeros of the given polynomial function. Do # 28, page 264.