write a polynomial function y of least degree that has rational coefficients, a leading coefficient of 1, and the zeros -2, 1, 4+i

Question

i need help fast for a math test. I am struggling

Answer 1

Hi Megan,

You know what a zero (also called a root) means? It means that (for polynomials of degree 3, anyway), in addition to the function having the value of zero at that x value, that there exists a factorization of the function into a product of  terms, one (or more) of which individually have the value zero at that x value. That is because, for any product (of terms, numbers or whatever), the result CAN'T be zero UNLESS one of the factors has the value zero (think about that, until you "get" it).

So, initially, write your function as y = (x+2)*(x-1)*(x-(4+i)). That equation just expresses that there are three factors which each COULD make the product zero, and that each of them only is present a single time in the product function (they could be present more times, e.g. as ((x+2)^2), but that would increase the degree of the product, which "least" forbids!). Expand this out, collect terms of the same order (power) in x, and you're done (if you need to, multiply or divide through everything by a constant so as to get that coefficient on x^3 to be 1 exactly).

--S.