Final answer:
To construct a polynomial function of degree 3 with zeros √5, -√5, and √7, we use the factors (x - √5), (x + √5), and (x - √7), which results in the function f(x) = x³ - √7x² - 5x + 5√7.
Step-by-step explanation:
The question asks for a polynomial function of degree 3 with given zeros: √5, -√5, and √7. To find such a polynomial, we can use the fact that if a number 'r' is a zero of a polynomial, then (x - r) is a factor of that polynomial.
Since the zeros include square roots, we remember that x² = √x. Therefore, the factors corresponding to the given zeros will be (x - √5), (x + √5), and (x - √7). Multiplying these factors together gives us the polynomial function:
f(x) = (x - √5)(x + √5)(x - √7)
Expanding these factors, we use the difference of squares for the first two factors, which simplifies to x² - 5. Then we multiply by the third factor, resulting in the cubic polynomial:
f(x) = (x² - 5)(x - √7) = x³ - √7x² - 5x + 5√7
This is our required polynomial function with the degree of 3 and the given zeros.