Write the cubic polynomial function f(x) in expanded form with zeros −4, −2, and 1, given that f(−1)=−24 .

Question

Write the cubic polynomial function f(x) in expanded form with zeros-4,-2 and 1, given that f(-1) =- 24.

f(x)=

Answer 1

If f(-1) = -24, that means that one point on the graph is (-1,-24).

-24 = a(-1 - (-4))(-1 - (-2))(-1 - 1)

24 = a(5)(3)(-2)

a = 24/-30

a = -4/5

f(x) = (-4/5)(x + 4)(x + 2)(x - 1)

Answer 2

Given zeros of -4, -2, and 1, the cubic polynomial must be of the form:

f(x) = a(x + 4)(x + 2)(x - 1)

Now, we need to find the value of a. Since f(-1) = -24, we have:

-24 = a(3)(-2)

a = 4

Therefore, the cubic polynomial is:

f(x) = 4(x + 4)(x + 2)(x - 1) = 4(x² + 6x + 8)(x - 1) = 4x³ - 20x² + 8x - 32

Answer 3

A cubic polynomial will have 3 zeros. Sometimes there may be two of them that are complex (real number plus imaginary) but in this case they are all three real numbers (−4, −2, and 1). We can write the polynomial in factored form using the binomial associated with each zero. For -4, the binomial is (x - (-4)) or (x + 4). For -2, the binomial is (x - (-2)) or (x + 2), and for 1, it is (x - 1).

The cubic polynomial is then (x + 4)(x + 2)(x - 1) HOWEVER, there can be a multiplier "a" in front of these like this:

f(x) = a(x + 4)(x + 2)(x - 1)

To find "a", we use the fact that f(−1)=−24

-24 = a((-1) + 4)((-1) + 2)((-1) - 1)

-24 = a(3)(1)(-2)

-24 = -6a

a = 4

So the cubic polynomial is f(x) = 4(x + 4)(x + 2)(x - 1) BUT, we are to write it in expanded form so we need to multiply it out.

Let's start with (x + 4)(x + 2). Using FOIL (first, outside, inside, last) we get x² + 2x + 4x + 8 = x² + 6x + 8

Now multiply that and (x - 1):. We can use a generic rectangle for that:

So the product is x³ + 6x² + 8x - x² - 6x - 8 = x³ + 5x² + 2x - 8

Now, multiply by the "4" to get the final answer:

f(x) = 4x³ + 20x² + 8x - 32