Sure, let's go through the steps to find the equation of the cubic polynomial function whose graph has zeroes at 2, 3, and 5.
Given that the zeroes of the polynomial are 2, 3, and 5, we can express the polynomial function as:
[tex]\[ f(x) = (x - 2)(x - 3)(x - 5) \][/tex]
We need to simplify this expression. Let's start by multiplying the first two factors:
[tex]\[ (x - 2)(x - 3) \][/tex]
Multiplying these two binomials:
[tex]\[ (x - 2)(x - 3) = x(x - 3) - 2(x - 3) \][/tex]
[tex]\[ = x^2 - 3x - 2x + 6 \][/tex]
[tex]\[ = x^2 - 5x + 6 \][/tex]
Next, we need to multiply this result by the third factor [tex]\( (x - 5) \)[/tex]:
[tex]\[ (x^2 - 5x + 6)(x - 5) \][/tex]
Multiplying these terms carefully, we get:
[tex]\[ (x^2 - 5x + 6)(x - 5) = x^2(x - 5) - 5x(x - 5) + 6(x - 5) \][/tex]
[tex]\[ = x^3 - 5x^2 - 5x^2 + 25x + 6x - 30 \][/tex]
Now, combine like terms:
[tex]\[ = x^3 - 10x^2 + 31x - 30 \][/tex]
So, the simplified polynomial function is:
[tex]\[ f(x) = x^3 - 10x^2 + 31x - 30 \][/tex]
Therefore, the equation of the polynomial function is:
[tex]\[ f(x) = x^3 - 10x^2 + 31x - 30 \][/tex]
This matches with the given answer, ensuring that we have followed the correct steps.
