Find a polynomial of degree 3 that has zeros 1,-2, and 3 and with the coefficient of x^2 is 5.
If k is a zero of a function, then (x-k) is a factor, so
If 1 is a zero (x-1) is a factor
If -2 is a zero then (x+2) is a factor
If 3 is a zero then (x-3) is a factor.
Then a third degree polynomial with these zeros is:
In order for the coefficient on the `x^2` term to be 5, we must multiply through by `-5/2` getting
The polynomial of degree 3 has zeros 1, -2 and 3. The polynomial can be written as (x - 1)(x + 2)(x - 3)
=> (x^2 - x + 2x - 2)(x - 3)
=> (x^2 + x - 2)(x - 3)
=> x^3 + x^2 - 2x - 3x^2 - 3x + 6
=> x^3 - 2x^2 - 5x + 6
The coefficient of x^2 is 5,
=> (-5/2)*x^3 + 5x^2 + (25/2)*x - 15
The required polynomial is the polynomial is (-5/2)*x^3 + 5x^2 + (25/2)*x - 15