Question

A cubic polynomial function f has a leading coefficient of 2 and a constant term of −5. When f(1) = 0 and f(2) = 3, what is f(−5)? A) -21 B) 31 C) -31 D) 21

Answer 1

The cubic polynomial function f can be found by using the values of f(1) = 0 and f(2) = 3. By solving these equations simultaneously, the function can be written as f(x) = 2x³ - 6x² + 9x - 5. Substituting x = -5 into the equation, we find that f(-5) = -31.

Step-by-step explanation:

To find f(-5), we need to find the equation of the cubic polynomial function f. Given that the function has a leading coefficient of 2 and a constant term of -5, we can write the equation as:

f(x) = 2x³ + bx² + cx - 5

Using the two given values f(1) = 0 and f(2) = 3, we can substitute these values in the equation to solve for b and c.

By substituting f(1) = 0, we get:
2(1)³ + b(1)² + c(1) - 5 = 0
2 + b + c - 5 = 0
b + c = 3

By substituting f(2) = 3, we get:
2(2)³ + b(2)² + c(2) - 5 = 3
16 + 4b + 2c - 5 = 3
4b + 2c = -8

Solving these two equations simultaneously, we find that b = -6 and c = 9.

Substituting these values into the equation f(x), we get:

f(x) = 2x³ - 6x² + 9x - 5

Finally, substituting x = -5, we can find f(-5):

f(-5) = 2(-5)³ - 6(-5)² + 9(-5) - 5 = -31

Answer 2

Given that the leading coefficient is 2, this means a = 2

Also, given that the constant term is -5, this means d = -5.

We are told that f(1) = 0 and f(2) = 3

Let's use this information to set up two equations:

1. f(1) = a(1)^3 + b(1)^2 + c(1) + d = 0

2. f(2) = a(2)^3 + b(2)^2 + c(2) + d = 3

Using a = 2 and d = -5, we can write these equations:

1. 2 + b + c - 5 = 0

2. 16 + 4b + 2c - 5 = 3

Now, solving the system of equations, we get \( b = -9 \) and \( c = 14 \).

substituting these values into the cubic polynomial f(x):

f(x) = 2x^3 - 9x^2 + 14x - 5

Now, find f(-5):

f(-5) = 2(-5)^3 - 9(-5)^2 + 14(-5) - 5

f(-5) = -21

So, the correct answer is -21, which corresponds to option A.