Question

Write the cubic polynomial function f(x)in expanded form with zeros −6, −4,and −1,given that f(−2)=−32.

Answer 1

Answer:

$f(x)=4x^3+44x^2+136x+96$

Step-by-step explanation:

Polynomials

It's possible to build a polynomial function by knowing its zeros and leading coefficient.

Given the zeros of a third-degree polynomial: x=x1, x=x2, and x=x3, the function is:

$f(x)=a(x-x_1)(x-x_2)(x-x_3)$

Where a is the leading coefficient.

We are given the zeros -6, -4, and -1, thus:

$f(x)=a(x+6)(x+4)(x+1)$

The value of a can be calculated by substituting the point (-2,-32):

$a(-2+6)(-2+4)(-2+1)=-32$

Calculating:

$a(4)(2)(-1)=-32$

$-8a=-32$

Dividing by -8:

$a = -32/(-8) =4$

a = 4.

The polynomial is now complete:

$f(x)=4(x+6)(x+4)(x+1)$

Operating:

$f(x)=4(x^2+10x+24)(x+1)$

$f(x)=4(x^3+11x^2+34x+24)$

$\mathbf{f(x)=4x^3+44x^2+136x+96}$