Sure, to find the polynomial f(x) of degree 4 that has zeros of 2, 6, -9, and -8, we follow four basic steps.
The zeros of a polynomial are the x-values that make the polynomial equal to zero. Meaning, if we substitute any of these zeros into the polynomial, it yields 0 as the result. Knowing this, we can construct a polynomial from the given zeros.
Step 1: Start by noting down the zeros or roots of the polynomial:
2, 6, -9, -8
Step 2: The zeros of the polynomial can help us to construct the polynomial f(x) as the product of linear factors. To construct these factors, each zero transforms into the form (x - zero).
For the zero of 2, the factor will be (x - 2).
For the zero of 6, the factor will be (x - 6).
For the zero of -9, the factor will transform into (x - (-9)) = (x + 9).
For the zero of -8, the factor will transform into (x - (-8)) = (x + 8).
Step 3: The next step is to multiply these linear factors to create the polynomial. The formula to construct the polynomial f(x) is:
f(x) = (x - zero_1) * (x - zero_2) * ... * (x - zero_n)
Thus, substituting the values, you get:
f(x) = (x - 2) * (x - 6) * (x + 9) * (x + 8).
Step 4: Simplifying, you will find that f(x) = (x - 2)(x - 6)(x + 9)(x + 8) = (--8 + x)(--9 + x)(-2 + x)(-6 + x).
So, the polynomial f(x) of degree 4 that has the zeros of 2, 6, -9, and -8 is f(x) = (--8 + x)(--9 + x)(-2 + x)(-6 + x). Here, the '--' symbol in '--8' and '--9' is used to represent a double negative, making -8 and -9 positive.