Zeroes, also called roots, of a function are input values that produce an output of zero. The zeroes of a polynomial function P(x) can be found by factoring P(x) into its linear factors. A polynomial of degree n will have n complex zeroes. A zero is considered a multiple zero if its linear factor repeats, increasing the zero's multiplicity. The Rational Zero Theorem states that if a polynomial with rational coefficients has rational zeroes, they must be integers that are factors of the constant term divided by factors of the leading coefficient.

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) = 𝑥2
+ 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) = 𝑥2
+ 9x + 18
P(x) = (x + 6)(x + 3)
2. P(x) = 𝑥2 - 16
P(x) = (x -4)(x + 4)
3. P(x) = 2𝑥2
- 5x - 3
P(x) = (2x + 1) (x – 3)
4. P(x) = 15𝑥2
- x - 2
P(x) = (5x – 2) (3x + 1)
Zeroes are -6, -3
Zeroes are 4, -4
Zeroes are − Τ
1 2 , 3
Zeroes are
Τ
2 5 , − Τ
1 3

5. Fundamental Theorem on Algebra
A polynomial function P(x) of degree n has
exactly n complex zeroes.
Examples
1. P(x) = 𝒙𝟒
+ 𝟐𝒙𝟑
- 5𝒙𝟐
+ 2 has 4 complex
zeroes
2. P(x) = 𝟑𝒙𝟑𝟏
+ 2𝒙𝟐𝟏
- 𝟐𝒙𝟏𝟏
+ 7x has 31
complex zeroes

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

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

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. 𝑥3 + 2𝑥2 - 8x + 10
2. 𝑥4
- 2𝑥2
- 5x + 6
3. 𝑥5 - 2𝑥4 - 3x - 18
±1, ±2, ±5, ±10
±1, ±2, ±𝟑, ±𝟔
±1, ±2, ±𝟑, ±𝟔, ±𝟗, ±𝟏𝟖

10. Rational Zero Theorem
If the Polynomial P(x) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛 −1𝑥𝑛−1 + . . . +
𝑎1x + 𝑎0 has rational roots then the roots must
be in the form ± 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑝
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑞
Where:
P = factors of constant term
Q = factors of leading coefficient

11. Examples:
1.2x2 – 5x -3
2.15x4 – x3 – 17 x2 + x + 2