Answer:
A least degree polynomial, having rational coefficients and a leading coefficient of 2, with,-4, 0, 2, and 4 as the zeros of the polynomial is;
f(x) = 2·x⁴ - 4·x³ - 32·x² + 64·x
Step-by-step explanation:
The given parameters of the polynomial are;
The leading coefficient of the polynomial = 2
The zeros of the polynomial = -4, 0, 2, 4
We note that zeros of -4, and 4 gives a factor of the form, (x² - 4²)
For a zero of the polynomial equal to 0, one of the factors of the polynomial is equal to 'x'
To have a leading coefficient of 2, we can add '2' as a factor of the polynomial
Therefore, we can have the factors of the polynomial as follows;
(x² - 4²)·2·x×(x - 2) = 0
From the above equation, using a graphing calculator, we get the following possible polynomial;
(x² - 4²)·2·x×(x - 2) = 2·x⁴ - 4·x³ - 32·x² + 64·x = 0
Therefore, a polynomial, function of least degree that has rational coefficients, a leading coefficient of 2,and the zeros, -4, 0, 2, and 4 is f(x) = 2·x⁴ - 4·x³ - 32·x² + 64·x.
