A cubic polynomial function with given zeros can be represented in its standard form. For the zeros 1, 2, and 2, we can express the function as follows:
If a polynomial has a zero at a point, it means that point is a root of the polynomial. Therefore, a polynomial with zeros at 1 and 2 (with 2 being a repeated root) can be initially written in its factored form:
f(x) = (x – 1)(x – 2)(x – 2)
We can simplify this to:
f(x) = (x – 1)(x – 2)^2
Next, we will expand this expression. First, we need to expand the squared term:
(x – 2)^2 = x^2 – 4x + 4
Now, we substitute this back into our function:
f(x) = (x – 1)(x^2 – 4x + 4)
To further simplify, we will distribute (x – 1) across the quadratic term:
f(x) = x(x^2 – 4x + 4) – 1(x^2 – 4x + 4)
This yields:
f(x) = x^3 – 4x^2 + 4x – (x^2 – 4x + 4)
Combining like terms results in:
f(x) = x^3 – 5x^2 + 8x – 4
Thus, the cubic polynomial function in standard form with zeros 1, 2, and 2 is:
f(x) = x^3 – 5x^2 + 8x – 4