Question

Which of the following functions best represents the graph?

f(x) = x3 + x2 − 4x − 4
f(x) = x3 + 4x2 − x − 4
f(x) = x3 + 3x2 − 4x − 12
f(x) = x3 + 2x2 − 4x − 8

Answer 1

Choice A: f(x) = x^3 + x^2 − 4x − 4

Explanation:

Here's a great and simple answer.

Ok first we need to take the x intercepts to solve.

If we look at the graph we see the x ints are -2,+1 and +2.

To solve we need to put them into factor form

= (x-2) (x+2) and (x+1)

Simplify: (x-2) (x+2) = (x^2-4) and (x+1)

Now we take (x^2-4) and (x+1) and multiply them to find our answer

(x^2-4) (x+1)

= x^2(x) and x^2(1) = x^3 and x^2.

now the other: -4(x) and -4(+1) = -4x and -4

We have nothing common here so we just join them

= x^3 + x^2 - 4x - 4, and that is the same as choice A.

Answer 2

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

Explanation:

From the graph, the x-intercepts are;

`x=-2`

`x=-1`

`x=2`

These are root of the polynomial function represented by the given graph.

By the remainder theorem;

`f(-2)=0,f(-1)=0,f(2)=0`

According to the factor theorem, if
`(x-a)` is a factor of
`f(x)`, then
`f(a)=0`

This implies that;

`(x+2),(x+1),(x-2)` are factors of the required function.

Hence;
`f(x)=(x+1)(x-2)(x+2)`

We expand using difference of two squares to obtain;

`f(x)=(x+1)(x^2-4)`

We expand using the distributive property to get;

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

Rewrite in standard form to obtain;

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