Let's break down the question to find the zeros of the function [tex]\( y = x^2 (x^2 - 6x + 9) \)[/tex].
1. Identify the function and simplify it:
The function given is [tex]\( y = x^2 (x^2 - 6x + 9) \)[/tex]. To simplify the expression inside the parentheses, we notice that:
[tex]\[
x^2 - 6x + 9 = (x - 3)^2
\][/tex]
Therefore, the function simplifies to:
[tex]\[
y = x^2 (x - 3)^2
\][/tex]
2. Identify the zeros of the function:
The zeros of the function occur where the function evaluates to zero. This can be achieved by setting each factor equal to zero:
[tex]\[
x^2 = 0 \quad \text{or} \quad (x - 3)^2 = 0
\][/tex]
Solving these equations:
[tex]\[
x^2 = 0 \implies x = 0
\][/tex]
[tex]\[
(x - 3)^2 = 0 \implies x = 3
\][/tex]
Therefore, the zeros of the function are [tex]\( x = 0 \)[/tex] and [tex]\( x = 3 \)[/tex].
3. Count the distinct real zeros:
The zeros we found are [tex]\( x = 0 \)[/tex] and [tex]\( x = 3 \)[/tex], both of which are real numbers.
Thus, the function has 2 distinct real zeros.
4. Determine the number of complex zeros:
Since all the zeros we found are real numbers, there are no complex zeros.
Thus, the graph of the function [tex]\( y = x^2 (x^2 - 6x + 9) \)[/tex] has zeros of [tex]\( 0 \)[/tex] and [tex]\( 3 \)[/tex], so the function has 2 distinct real zeros and 0 complex zeros.
### Fill-in the blanks:
- The graph of the function [tex]\( y = x^2 (x^2 - 6 x + 9) \)[/tex] has zeros of 0, 3 , so the function has 2 distinct real zeros and 0 complex zeros.
