Final answer:
To find a degree 3 polynomial with integer coefficients and given zeros, each zero is converted into a polynomial factor, and these factors are multiplied together to form the polynomial.
Step-by-step explanation:
To write a degree 3 polynomial with integer coefficients that has zeros of −5/3, 4/5, and 3, we'll use the fact that if a polynomial has a zero at x = a, then (x − a) is a factor of the polynomial. First, start by rewriting the zeros in the form of factors:
The factor for the zero −5/3 is 3x + 5 because when 3x + 5 = 0, x = −5/3. Similarly, the factor for the zero of 4/5 is 5x − 4, and for the zero of 3 the factor is x − 3.
We can now construct the polynomial by multiplying these factors:
f(x) = (3x + 5)(5x − 4)(x − 3).
To ensure that all coefficients are integers, we multiply the factors as they are, without any need for further adjustment.
Calculating the product of these factors involves using the distributive property (foil and area methods) to expand the expression into a polynomial:
f(x) = (3x + 5)(5x^2 − 15x − 4x + 12)
f(x) = (3x + 5)(5x^2 − 19x + 12)
f(x) = 15x^3 − 57x^2 + 36x + 25x^2 − 95x + 60
f(x) = 15x^3 − 32x^2 − 59x + 60
This polynomial meets the conditions specified in the question and has integer coefficients.