What is the value of f(x) divided by g(x) when f(x) is 2x^3 + x^2 – 16x – 15 and g(x) is x – 3?

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Finding f(x) over g(x)

To find the value of f(x) divided by g(x), we first need to define the functions:

  • f(x) = 2x3 + x2 – 16x – 15
  • g(x) = x – 3

We will calculate f(x) and g(x) separately and then perform the division.

Step 1: Calculate f(x)

Let’s find f(x):

For simplicity, let’s plug in a value for x. We’ll test for x = 3:

        f(3) = 2(3)3 + (3)2 - 16(3) - 15
             = 2(27) + 9 - 48 - 15
             = 54 + 9 - 48 - 15
             = 0

Step 2: Calculate g(x)

Now, let’s calculate g(x) for the same value:

        g(3) = 3 - 3 = 0

Step 3: Evaluate f(x) / g(x)

At this point, we observe:

  • f(3) = 0
  • g(3) = 0

When both the numerator and denominator equal zero, we have an indeterminate form. To further analyze this, we might consider using polynomial long division or factoring out common terms.

Factorization of f(x)

We factor f(x) to see if x – 3 is a factor:

        f(x) = (x - 3)(2x2 + 7x + 5)

Final Result

Now we can write:

        f(x) / g(x) = (x - 3)(2x2 + 7x + 5) / (x - 3)
                     = 2x2 + 7x + 5, for x ≠ 3

Thus, the result of f(x) / g(x) can be simplified to:

2x2 + 7x + 5, as long as x is not equal to 3.

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