SWBAT to find real zeros of polynomials using the Rational Zero
Test ( Section 3-5)
Warm up
Find the zeros of the following polynomials
1)
f(x) = x 2 - 4
2)
f(x) = x 3  1
3)
f(x) = x 2 + 1
SWBAT to find real zeros of polynomials using the Rational Zero
Test (Lesson 16 - Section 3-5)
Recall:
We were able to identify the real zeros of a polynomial function
because either the polynomial function was in factored form or it
could be easily factored.
How do we find the real zeros of a polynomial function if it is not
factored or cannot be easily factored?
Theorem: Number of Real Zeros
A polynomial function of degree n, n ≥ 1, has at most n real zeros.
The Rational Zero Test
Let f be a polynomial function of degree 1 or higher of the form
f(x) = anxn + an-1xn-1 + …..+a1x + a0
a0 ≠ 0, an ≠ 0.
p
, in lowest terms, is a
q
rational zero of f, then p must be a factor of a0, and q must be a
factor of an.
Where each coefficient is an integer. If
SWBAT to find real zeros of polynomials using the Rational Zero
Test (Lesson 16 - Section 3-5)
Example 1) Listing the Potential Rational Zeros of
Example 2) Find the rational zeros of
Write f in factored form.
Example 3) Find the real zeros of
Write f in factored form.
f ( x)  2 x 3  11x 2  7 x  6
f ( x)  2 x 3  3 x 2  8 x  3 .
f ( x)  x 5  x 4  4 x 3  8x 2  32x  48
Steps for Finding the Real Zeros of a Polynomial Function
1) Use the degree of the polynomial to determine the maximum
number of zeros.
2) If the polynomial has integer coefficients, use the Rational
Zero Test Theorem to identify those rational numbers that
potentially can be zeros.
3) Using a graphing utility, graph the polynomial function.
4) Use the Factor Theorem to determine if the potential rational
zero is a zero. If it is, use synthetic division or long division
to factor the polynomial function. Each time that a zero (and
thus a factor) is found, repeat step 4 on the depressed
equation. In attempting to find zeros, remember to use (if
possible) the factoring techniques that you already know.