Zero Theorem and Rational Roots Presentation

Credits to my professor

2. Zero or Root of a function
In mathematics, a zero of a function,
also sometimes called a root, is an
input value(complex number) that
produces an output of zero(0).

3. Examples:
P(x) = ??????
2
+ 7x + 10
P(x) = (x + 2)(x+5) –factored form
therefore the zeroes are -2 and -5

4. What are the zeroes of the following functions.
1. P(x) = ??????
2
+ 9x + 18
P(x) = (x + 6)(x + 3)
2. P(x) = ??????
2
-16
P(x) = (x -4)(x + 4)
3. P(x) = 2??????
2
-5x -3
P(x) = (2x + 1) (x –3)
4. P(x) = 15??????
2
-x -2
P(x) = (5x –2) (3x + 1)
Zeroes are -6, -3
Zeroes are 4, -4
Zeroes are −Τ12,3
Zeroes are
Τ25,−Τ13

5. Fundamental Theorem on Algebra
A polynomial function P(x) of degree n has
exactly n complex zeroes.
Examples
1. P(x) = ??????
�
+ �??????
�
-5??????
�
+ 2 has 4complex
zeroes
2. P(x) = �??????
��
+ 2??????
��
-�??????
��
+ 7x has 31
complex zeroes

6. Multiple Zeroes of a function
If a polynomial P(x) has (x –r) occurring as a
factor exactly ktimes, then ris a zero of
multiplicityof k of the polynomial function y = P(x).

7. Examples:
Find the zeroes of each of the following
functions
1. P(x) = (??????+2)
3
(??????−3)
2
(x + 1)
2. P(x) = (3x−1)
2
(x+2)
2
(x-1).
3. P(x) = (??????+5)
3
(??????−7)
2
(2x -3)
-2, -2, -2, 3, 3, -1
Τ13,Τ13,−2,−2,1
-5, -5, -5, 7, 7, Τ32

8. Finding an Integral Zeros of
Polynomial Function
Let P(x) be a polynomial function in x
with integral coefficients. Then the only
possible integral zeroes are the divisors of
the constant term

9. Examples:
Find the possible integral zeroes of each of
the following functions.
Possible zeroes
1. ??????
3
+ 2??????
2
-8x + 10
2. ??????
4
-2??????
2
-5x + 6
3. ??????
5
-2??????
4
-3x -18
±1, ±2, ±5, ±10
±1, ±2, ±�, ±??????
±1, ±2, ±�, ±??????,±�,±��

10. Rational Zero Theorem
If the Polynomial P(x) = ??????
????????????
??????
+ ??????
??????−1??????
??????−1
+ . . . +
??????
1x + ??????
0has rational roots then the roots must
be in the form ±
??????????????????�����??????�
??????????????????�����??????�
Where:
P = factors of constant term
Q = factors of leading coefficient

11. Examples:
1.2x
2
–5x -3
2.15x
4
–x
3
–17 x
2
+ x + 2