find the polynomial function that has integer coefficients, a leading coefficient of 1, and the least degree for the given zeros 2 and 3-√7

Answer 1

The zeros of a function occur where the function's value is equal to zero. This can be easily seen in equations like y = x² - 4, which has zeros of -2 and 2. We can calculate these without graphing by rewriting the equation as y = (x - 2)(x + 2). The method for finding zeros in this way is to determine what number can replace the variable to result in 0.

We can also reverse engineer the aforementioned process. With this problem, we are given that the leading coefficient (what the first term is multiplied by) is 1 and the zeros are 2 and 3 - √7. Writing this as a function, we get f(x) = (x - 2)(x - (3 - √7). If the question requires it, you can multiply this out using FOIL and simplify to x² - 5x + x√7 + 6 - 2√7. All of these coefficients are integers (whole numbers), so the equation satisfies all requirements (from what I can tell).

Here is the line-by-line process of how I did this:

(x - 2)(x - (3 - √7))

x² - x(3 - √7) - 2x + 2(3 - √7)

x² - 3x + x√7 - 2x + 6 - 2√7

x² - 5x + x√7 + 6 - 2√7

Please let me know if I did anything wrong. Hope that helps!