Certainly! Let's simplify the polynomial function step-by-step.
Given the polynomial function with zeroes at [tex]\(2\)[/tex], [tex]\(3\)[/tex], and [tex]\(5\)[/tex], it can be written as:
[tex]\[ f(x) = (x - 2)(x - 3)(x - 5) \][/tex]
To simplify the right side, we need to first expand the factors step-by-step.
### Step 1: Expand [tex]\((x - 2)(x - 3)\)[/tex]
We start by expanding the first two factors:
[tex]\[ (x - 2)(x - 3) \][/tex]
Using the distributive property (also known as FOIL method for binomials):
[tex]\[ (x - 2)(x - 3) = x \cdot x + x \cdot (-3) - 2 \cdot x - 2 \cdot (-3) \][/tex]
[tex]\[ = x^2 - 3x - 2x + 6 \][/tex]
[tex]\[ = x^2 - 5x + 6 \][/tex]
Now we have:
[tex]\[ f(x) = (x^2 - 5x + 6)(x - 5) \][/tex]
### Step 2: Expand [tex]\((x^2 - 5x + 6)(x - 5)\)[/tex]
Next, we multiply this result by the remaining factor [tex]\( (x - 5) \)[/tex]:
[tex]\[ (x^2 - 5x + 6)(x - 5) \][/tex]
Using the distributive property again, we distribute each term in [tex]\( (x^2 - 5x + 6) \)[/tex] across [tex]\( (x - 5) \)[/tex]:
[tex]\[ = (x^2 - 5x + 6) \cdot x + (x^2 - 5x + 6) \cdot (-5) \][/tex]
Expand each term separately:
[tex]\[ = x^3 - 5x^2 + 6x - 5x^2 + 25x - 30 \][/tex]
Now combine like terms:
[tex]\[ = x^3 - 10x^2 + 31x - 30 \][/tex]
Therefore, the simplified form of the polynomial function is:
[tex]\[ f(x) = x^3 - 10x^2 + 31x - 30 \][/tex]
So, the correct simplified equation is:
[tex]\[ \boxed{f(x) = x^3 - 10x^2 + 31x - 30} \][/tex]
