Question

Which statements are true about the polynomial function?

f(x)=x^3−x^2−4x+4

Select each correct answer.
f(−1)=0
f(x)=0 when x=−1
(x−1) is a factor of f(x)
f(x) divided by (x−1) has a remainder of 0

Answer 1

Explanation:

Let's evaluate each of the given statements one by one.

1. f(−1)=0

To determine if this statement is true, plug x = -1 into the function and calculate the value:

`f(-1) = (-1)^3 - (-1)^2 - 4(-1) + 4`

`f(-1) = -1 - 1 + 4 + 4`

`f(-1) = -2 + 8`

`f(-1) = 6`

So f(−1) does not equal 0, which means the statement "f(−1)=0" is false.

2. f(x)=0 when x=−1

This is essentially the same statement as the first one just worded differently. We've already determined that f(-1) is not equal to 0, so this statement is also false.

3. (x−1) is a factor of f(x)

If
`(x - 1)` is indeed a factor of
`f(x)`, then the remainder when
`f(x)` is divided by
`(x - 1)` should be 0. This can be checked using synthetic division or by applying the remainder theorem which states that if a polynomial
`f(x)` is divided by
`(x - c)`, the remainder is
`f(c)`. Let's check
`f(1)`:

`f(1) = 1^3 - 1^2 - 4(1) + 4`

`f(1) = 1 - 1 - 4 + 4`

`f(1) = 0`

Since
`f(1) = 0`,
`(x - 1)` is indeed a factor of
`f(x)`, so this statement is true.

4. f(x) divided by (x−1) has a remainder of 0

From the previous statement, we know
`(x - 1)` is a factor of
`f(x)`, so dividing
`f(x)` by
`(x - 1)` should indeed give a remainder of 0. Therefore, this statement is also true.

In summary, only the third and fourth statements are true:

- (x−1) is a factor of f(x)

- f(x) divided by (x−1) has a remainder of 0