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Transcript of 0 0 0 0 0 0 0 0 Finding Rational Zeros. Zeros = Solutions = Roots = x-intercepts Find all zeros of x...
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Finding Rational Zeros
Zeros = Solutions = Roots = x-intercepts
Find all zeros of x2 –10x + 24
You just factor and set each factor to zero. (You knew this already!)
(x – 12)(x + 2) = 0 x = 12, -2
You could also graph and see where it crosses the x-axis (x-intercepts) (You knew this already too!)
Refresh. What is a Rational Number?
A number that can be written as the ratio of two integers.
Examples: 31
197
15311
It can also be an ending or repeating decimal.
Examples: 6.21212121… 9.58310562 0.3333333…
Rational Zero Theorem
If f(x) = anxn + … + a1x + a0 (it’s a polynomial)
and the polynomial has integer coefficients, then
EVERY rational zero of f has the following form:
qp
factor of the constant term . factor of the leading coefficient
=
“p over q”
Find the rational zeros of f(x) = x3 + 2x2 – 11x – 12
p is all of the factors of the constant term.
1, 3,2, 4, 6, 12
q is all of the factors of the leading coefficient.
This one is easy because the leading coefficient is 1 !
The only factor is: 1
qp
= 1, 3,2, 4, 6, 12
Using qp
Still finding the rational zeros of f(x) = x3 + 2x2 – 11x – 12
1, 3,2, 4, 6, 12 Possible Zeros:
Do synthetic division until you find a zero.
x 3 + 2 x
2 + (-11) x + (–12)
1 3
1
-8
3
-20
-8
1 2 –11 - 121
k-value
1 •
Remainder is not z
ero so
1 is not a
zero to
this f
unction.
Test x = -1.
Keep trying Possible Zeros
x 3 + 2 x
2 + (-11) x + (–12)
1 1
-1
-12 0
12
1 2 –11 - 12-1
k-value
-1 •
Remainder IS ze
ro so
-1 IS a ze
ro to th
is functi
on!
Since -1 is a zero of f, then the result is a factor. (x2 + x – 12)
This is factorable into: (x + 4)(x – 3).
The zeros are: -1 (original zero), -4, 3
-1
The Nightmare Example
Find the rational zeros of f(x) = 10x4 - 3x3 - 29x2 + 5x + 12
qp
= 1, 3,2, 4, 6, 12
1 11 1 1 1
1, 3,2, 4, 6, 12
2 22 2 2 2
1, 3,2, 4, 6, 12
5 55 5 5 5
1, 3,2, 4, 6, 12
10 1010 10 10 10
The Nightmare Example (cont’d)
Finding the zeros of f(x) = 10x4 - 3x3 - 29x2 + 5x + 12
With so many possible zeros, it’s worth our time to get a ballpark answer by graphing the polynomial on the calculator.
10 -18
-15
-2
27
8
3 -12
0
10 -3 -29 5 12 -3/2 •
We found the 1st Zero!
What do you need to remember?
If f(x) = anxn + … + a1x + a0 (it’s a polynomial)
and the polynomial has integer coefficients, then
EVERY rational zero of f has the following form:
qp
factor of the constant term . factor of the leading coefficient
=
Rational Zero Theorem
Be able to list all possible rational zeros.Then let your calculator do the rest!