1. ## Zeros of polynomial

Given f(x) = x^2 (x-3)^3 (x+2)^2
1. How many zeros does the polynomial have?
2. What are the distinct zeros?
3. Give the multiplicity of each zero
4. Which zeros cross the x-axis?
5. Which zeros touch the x-axis?

2. Hello, oceanmd!

You are expected to know a few things . . .

Given: .$\displaystyle f(x) \:= \:x^2(x-3)^3(x+2)^2$

1. How many zeros does the polynomial have?
This is a seventh-degree polynomial . . . It will have seven zeros.
. . [They may not be distinct or real.]

2.What are the distinct zeros?
There are three distinct zeros: .$\displaystyle 0,\:3,\:-2$

3.Give the multiplicity of each zero.

The multiplicity of each factor is given by its exponent.

$\displaystyle x = 0$ has multiplicity 2
$\displaystyle x = 3$ has multiplicity 3.
$\displaystyle x = \text{-}2$ has multiplicity 2.

4. Which zeros cross the x-axis?
5. Which zeros touch the x-axis?

If the multiplcity is odd, the graph cross the x-axis.
If the multiplicity is even, the graph is tangent to the x-axis.

. . Got it?

3. ## Distinct and Real Zeros

Soroban, thank you for the explanation!!!

1. Whatever the degree of the polynomial is, it is the total number of zeros. (real and complex) Is this correct?
2. Does "distinct zeros" mean "real zeros" It is the same thing?
3. For this polynomial, there are three real zeros, does it mean that it has four complex zeros?

thank you

4. 1. Yes the degree is the amount of zeros.

2. No distinct is how many unique zeros. You can see in that problem that 3 is repeated 3 times, -2 twice, and zero would be a factor twice, but the two extra threes, and the extra -2, and 0 are not considered distinct because they are repeated.

3. No the factorization would be 0, 0, 3, 3, 3, -2, -2 which none are complex.