To solve the problem, we start by analyzing the function [tex]\( y = x^2 \left( x^2 - 6x + 9 \right) \)[/tex]. We need to find the zeros of the function.
Step 1: Factor the polynomial inside the parentheses:
[tex]\[ x^2 - 6x + 9 \][/tex]
This can be factored as:
[tex]\[ (x - 3)^2 \][/tex]
So the function becomes:
[tex]\[ y = x^2 (x - 3)^2 \][/tex]
Step 2: Determine when the function equals zero:
[tex]\[ y = 0 \][/tex]
This will happen when either factor equals zero:
[tex]\[ x^2 = 0 \][/tex]
or
[tex]\[ (x - 3)^2 = 0 \][/tex]
From [tex]\( x^2 = 0 \)[/tex]:
[tex]\[ x = 0 \][/tex]
From [tex]\( (x - 3)^2 = 0 \)[/tex]:
[tex]\[ x = 3 \][/tex]
Step 3: Identify the zeros:
The zeros are [tex]\( x = 0 \)[/tex] and [tex]\( x = 3 \)[/tex].
Step 4: Determine the distinct real zeros:
The distinct real zeros are [tex]\( x = 0 \)[/tex] and [tex]\( x = 3 \)[/tex].
Step 5: Count the number of distinct real zeros:
There are 2 distinct real zeros.
Thus, the function has:
- No complex zeros
- 2 distinct real zeros
So the correct selections for the drop-down menus are:
- Where it says 'complex zeros', select "no complex zeros"
- Where it says 'distinct real zeros', select "2 distinct real zeros"
The correctly filled statement should read:
The graph of the function [tex]\( y = x^2 (x^2 - 6x + 9) \)[/tex] has no complex zeros, so the function has 2 distinct real zeros and no complex zeros.
