Precal Help?

Question

Find a polynomial equation f(x) = 0satisfying the given conditions. (If not possible, enter IMPOSSIBLE.

Degree 3; the coefficient of x³ is 1; three roots are 3, −2, and 5

f(x)=??

Answer 1

The degree of the polynomial is 3.

Genral equation third degree polynomial ax³+ bx²+ cx +  d = 0

From factor therom f(c) = 0, x -c is factor of a polynomial.

Zeros of the function is 3, -2 and 5.

Therfore the factors are (x - 3),(x + 2) and (x - 5)

f(x) = ax³+ bx²+ cx +  d

f(x) = (x - 3)(x + 2)(x - 5)

     = [(x - 3)(x + 2)](x - 5)

     = [ x²+ 2x - 3x - 6](x - 5)

    = x³- 5x² + 2x² - 10x - 3x² + 15x - 6x + 30

    = x³ - 6x² - x + 30

Therfore, f(x) = x³ - 6x² - x + 30.