To determine the zeros of the function [tex]\( y = x^2 (x^2 - 6x + 9) \)[/tex], let's factorize the expression inside the parentheses:
First, simplify the function inside the parentheses:
[tex]\[ x^2 - 6x + 9 \][/tex]
Notice that [tex]\( x^2 - 6x + 9 \)[/tex] can be factored as a perfect square:
[tex]\[ x^2 - 6x + 9 = (x - 3)^2 \][/tex]
Thus, the original function can be rewritten as:
[tex]\[ y = x^2 (x - 3)^2 \][/tex]
To find the zeros of the function, set [tex]\( y = 0 \)[/tex]:
[tex]\[ x^2 (x - 3)^2 = 0 \][/tex]
This equation will be zero if any of the individual factors are zero:
[tex]\[ x^2 = 0 \quad \text{or} \quad (x - 3)^2 = 0 \][/tex]
Solving these, we get:
[tex]\[ x = 0 \quad \text{or} \quad x - 3 = 0 \][/tex]
[tex]\[ x = 0 \quad \text{or} \quad x = 3 \][/tex]
Therefore, the zeros of the function are:
[tex]\[ 0 \ \text{and} \ 3 \][/tex]
Next, count the number of distinct real zeros and complex zeros. Both 0 and 3 are real numbers and distinct from each other. There are no complex zeros.
So, the graph of the function [tex]\( y = x^2 (x^2 - 6x + 9) \)[/tex] has zeros of [tex]\(\{0, 3\}\)[/tex], so the function has [tex]\(\{2\}\)[/tex] distinct real zeros and [tex]\(\{0\}\)[/tex] complex zeros.
Therefore, the correct answer would be:
The graph of the function [tex]\(y = x^2(x^2 - 6x + 9)\)[/tex] has zeros of [tex]\(\{0, 3\}\)[/tex], so the function has [tex]\(2\)[/tex] distinct real zeros and [tex]\(0\)[/tex] complex zeros.
