Write the cubic polynomial function f(x) in expanded form with zeros −5, −2, and 3, given that f(−3)=36 .

Answer 1

If c is a zero of a polynomial function then (x - c) is a factor of the polynomial. Thus the factors of this polynomial are (x+5), (x + 2) and (x - 3).

An equation for the polynomial is f(x) = a(x + 5)(x + 2) (x -3)

Use that we are also given f(-3) = 36 in order to determine the value of a.

f(-3) = 36 = a(-3 + 5)(-3 + 2)(-3 -3)

36 = a(2)(-1)(-6)

36 = 12a

3 = a

Now, f(x) = 3(X + 5)(x + 2)(x - 3)

In expanded form, f(x) = 3(x + 5)(x² - x - 6)

f(x) = 3(x³ - x² - 6x + 5x² - 5x - 30)

f(x) = 3(x³ + 4x² - 11x - 30)

ANSWER: f(x) = 3x³ + 12x² - 33x - 90