When the Function f(x) is Divided by (x-2), the Quotient is x^2 6x 8 and the Remainder is -1

When we divide the function f(x) by (x-2), we get a quotient of x^2 6x 8 and a remainder of -1. This division can be written in a general form for polynomial division, where the dividend (f(x)) is equal to the divisor (x-2) times the quotient (x^2 6x 8), plus the remainder (-1). The process involves several steps and can be easily understood with some basic algebraic manipulation.

Step-by-Step Solution

Let's break down the steps to find the function f(x) in standard form:

General Formula for Polynomial Division

The general formula for polynomial division can be written as:

f(x) (x-2) (x^2 6x 8) - 1

Expanding the Expression

First, we expand the expression (x-2) (x^2 6x 8) using the distributive property (also known as the FOIL method for binomials, but extended to polynomials here).

Expanding (x-2) (x^2 6x 8) gives us:

x (x^2 6x 8) - 2 (x^2 6x 8)

Simplifying this, we get:

x^3 6x^2 8x - 2x^2 - 12x - 16

Combining like terms:

x^3 4x^2 - 4x - 16

Subtracting the Remainder

Now, we subtract the remainder, which is -1, from our expanded expression.

x^3 4x^2 - 4x - 16 - 1

This simplifies to:

f(x) x^3 4x^2 - 4x - 17

Verification

To verify our result, we use the division formula directly:

Let's write it out step-by-step:

1. Take the function f(x)

2. Divide by (x-2)