Question

Suppose that f(x) is a degree three polynomial function such that f(-5) = 0, f(0) = 0, f(3) = 0 If f(1) = 2.5, then what is the value of f(6.5)?

Answer 1

Based on the given information and calculations, the value of f(6.5) is approximately -50.62.

The given information tells us that the polynomial function f(x) is of degree three and has roots at x = -5, x = 0, and x = 3. This means that the equation of the function can be written as:

f(x) = a(x + 5)(x - 0)(x - 3)

We are also given that f(1) = 2.5, which means that when x = 1, the value of the function is 2.5. Plugging in these values into the equation above, we can solve for the value of a:

2.5 = a(1 + 5)(1 - 0)(1 - 3)

2.5 = a(6)(1)(-2)

2.5 = -12a

Dividing both sides of the equation by -12, we get:

a = -2.5/12

a = -0.20833 (rounded to 5 decimal places)

Now that we know the value of a, we can use it to find the value of f(6.5). Plugging x = 6.5 into the equation of the function, we get:

f(6.5) = -0.20833(6.5 + 5)(6.5 - 0)(6.5 - 3)

f(6.5) = -0.20833(11.5)(6.5)(3.5)

f(6.5) ≈ -50.62

Therefore, the value of f(6.5) is approximately -50.62.

Answer 2

The value of f(6.5) is -54.50

How to determine value of a function at a specified value of x.

Given that f(x) is a degree three polynomial with roots at x = -5, 0, and 3, let's express f(x) in factored form as

f(x) = a(x + 5)(x)(x - 3)

where (a) is a constant.

Given point (1, 2.5) on the function:

Substitute this point to find (a)

2.5 = a(1 + 5)(1)(1 - 3)

2.5 = a(6)(1)(-2)

2.5 = -12a

a = 2.5/12

= -5/24

Therefore, f(x) = -5/24(x + 5)(x)(x-3)

To determine f(6.5) by substitute x = 6.5 into the polynomial function.

f(6.5) = -5/24(6.5+5)(6.5)(6.5-3)

= -5/24(11.5)(6.5)(3.5)

= -54.50

The value of f(6.5) is -54.50