Download Irrational Zeros Rational Zero Theorem Synthetic & Long Division

Rational Zero Theorem
Synthetic & Long Division
Using Technology to Approximate
Zeros
Today you will look at finding zeros of higher
degree polynomials using the rational zero
theorem, synthetic division, and technology.
We will also review polynomial long division.
Try Factor by Grouping!
f ( x)  x  3x  4 x  12
3
2
f ( x)  x  2 x  11x  12
6
4
2
Factor by grouping worked fine for our first example, but not
for the second example so we need another way to find the
roots for our second equation.
We could just keep trying numbers until we find
something that works
OR
We can use the Rational Zero Theorem to
accomplish this more efficiently.
Getting to the Root of the (Polynomial)
Matter
You are often asked to find all the zeros (roots or x-intercepts) of
polynomials. To do this in the most efficient way, use the rational
zero test.
 First, there are some general concepts.
 When you FOIL a pair of quadratic binomials with leading coefficients of
one, notice that for y = (x – r1)(x – r2) , the constant in the trinomial
function is the product (r1 r2).
 When we to do the multiplication for y = (x – r1)(x – r2)(x – r3), the
constant in the polynomial function is the product (r1 r2 r3).
 That means that all rational zeros must be in the factor list for the
constant in a polynomial function when the leading coefficient is one.
Getting to the Root of the (Polynomial)
Matter
Rational Zero Theorem
If f(x) = anxn + . . . + a1x + a0 has integer
coefficients, then every rational zero of f has
the form:
p
--- =
q
factor of constant term a0 (c)
----------------------------------------factor of leading coefficient an (a)
Find all possible rational zeros of the
functions below using the Rational Zero Thm.
f ( x)  x 4  2 x 2  24
f ( x)  2 x5  x 2  16
f ( x)  6 x 4  3x 3  x  10
f ( x)  8 x  12 x  3
2
Let’s look at our example again:
f ( x)  x  2 x  11x  12
6
4
2
Let’s apply the Rational Zero Theorem to find all
possible rational zeros and then we can use Synthetic
Division to test those zeros.
Testing the possible rational zeros
using Synthetic Division.
Find the zeros of
f ( x)  2 x3  x 2  13x  6
 Find all possible rational zeros using Rational
Zero Theorem
 Use synthetic division to test rational zeros
 Factor or use quadratic formula (if you are left
with a quadratic function) to find remaining
zeros or use synthetic division again until you
are left with a quadratic function.
More Examples to Try
f ( x)  2 x  9 x  x  30
3
2
f ( x)  x  9 x  29 x  40 x  20
4
3
2
Use Polynomial Long Division to
Simplify
2x  x  2
2
2x  x 1
4
3
x  2 x  3x  4 x  6
2
x  2x 1
4
3
2
What happens if we use synthetic division and
none of the possible rational zeros work?
That means we must have Irrational Zeros.
Because they are irrational, we need to use
our graphing calculators to approximate
the zeros.
First, always graph the function. The
number of times the function crosses the
x-axis gives us the number of real zeros.
Zeros of Polynomial Functions Using
a Graphing Calculator
Approximate the real zeros of
1. Enter equation into Y1
4. Move cursor to left
of the zero – hit enter
2. Graph the equation
5. Move cursor to
right of zero – hit enter
f ( x)  x 4  2 x 3  x 2  2 x  2
3. Go to CALC: zero
6. Hit Enter one more
time to see the zero.
7. The coordinates of
the zero appear.
Repeat the process to find all other real zeros.
Zeros of Polynomial Functions Using
a Graphing Calculator
Try this one:
Approximate the real zeros of
f ( x)  6 x3  29 x 2  26 x  170
Assignment
A 1.9 Sect II & III
A 1.10
(Book Reference: Section 2.4 pgs. 214 – 223)
See yu tmrrw!