Question

Write a polynomial function in factored and standard form by'using the provided information:

f(x) is a third degree function; f(0) = - 12; roots are -2, 2, -3

Answer 1

The polynomial function, given roots -2, 2, -3 and f(0) = -12, is in factored form: f(x) = (x + 2)(x - 2)(x + 3). Multiplying the factors, the standard form is: f(x) = x^3 + 3x^2 - 4x - 12.

Step-by-step explanation:

To write a polynomial function in both factored and standard form using the given information that f(x) is a third degree function with roots -2, 2, and -3, and f(0) = -12, we start by constructing the factored form of the polynomial. Using the roots, the factored form is:

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

To find the value of 'a', we use the fact that f(0) = -12:

-12 = a(0 + 2)(0 - 2)(0 + 3)
-12 = a(2)(-2)(3)
-12 = -12a
a = 1

So the polynomial in factored form is:

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

To convert this into standard form, we multiply the factors:

f(x) = (x^2 - 4)(x + 3)
f(x) = x^3 + 3x^2 - 4x - 12

Thus, in standard form, the polynomial is:

f(x) = x^3 + 3x^2 - 4x - 12