what is the polyonimial function for -5, -1, 2

Question

what is the polynimial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros for -5, -1, 2

Answer 1

When writing a polynomial with given zeros, subtract each of the given zeros from x and then multiply the terms.

f(x) = [x - (-5)][x - (-1)](x - 2)              Write a polynomial function where each 0 is subtracted from x

f(x) = (x + 5)(x + 1)(x - 2)                    Subtracting negatives is adding positives

f(x) = (x² + x + 5x + 5)(x - 2)                FOIL the first 2 terms

f(x) = (x² + 6x + 5)(x - 2)                      Simplify

f(x) = x³ - 2x² + 6x² - 12x + 5x - 10        Multiply each term of the 1st parenthesis by each term of the 2nd

f(x) = x³ + 4x² - 7x - 10                         Simplify

 

Answer 2

Since you are give three zeros, we know we have a 3rd degree polynomial.   Recall that we factor a polynomial in the form of: 

(x-a)(x-b)(x-3) = 0

where a, b, and c are the zeros of the function.

Thus, your function factors as:

(x+5)(x+1)(x-2) = 0

Doing this in steps, we can distribute the first two quantities (FOIL) to get:

(x^2+6x+5)(x-2) = 0

Finally, we distribute the (x-2) to get

x^3 + 4x^2 - 7x - 10 = 0

You can plug in the zeros -5, -1, and 2 to check the answer.