Problem 11 - Precalculus: Functions and Graphs

Question

Use the remainder theorem to find \(f(c)\). $$f(x)=x^{4}-6 x^{2}+4 x-8 ; \quad c=-3$$

Step-by-Step Solution

Answer

The remainder is 7.

Step 1: State Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \(f(x)\) by \(x-c\) is \(f(c)\). Hence, to find \(f(c)\), we can use synthetic division to divide \(f(x)\) by \(x-c\) or directly substitute \(c\) into \(f(x)\).

Step 2: Substitute \(c\) into \(f(x)\)

To find \(f(-3)\), substitute \(-3\) for \(x\) in the polynomial: \[f(-3) = (-3)^4 - 6(-3)^2 + 4(-3) - 8.\]

Step 3: Calculate Each Term

Calculate each term separately: - \((-3)^4 = 81\), - \(-6(-3)^2 = -6 \times 9 = -54\), - \(4(-3) = -12\), - \(-8 = -8\).

Step 4: Sum the Results

Add all the calculated terms: \[f(-3) = 81 - 54 - 12 - 8.\] Simplifying gives: \[f(-3) = 81 - 54 = 27.\]\[27 - 12 = 15.\]\[15 - 8 = 7.\]

Step 5: Conclusion

Therefore, the remainder when \(f(x)\) is divided by \(x+3\) is \(f(-3)=7\).

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a divisor of the form \(x - c\). It's particularly useful because it reduces the amount of computation involved compared to long division. This technique is widely used in combination with the Remainder Theorem, making the process of finding the remainder much easier and quicker.

To perform synthetic division, you first write down the coefficients of the polynomial. In the case of \(f(x) = x^4 - 6x^2 + 4x - 8\), the coefficients are \([1, 0, -6, 4, -8]\). Notice that we include \(0\) for any missing terms, such as the \(x^3\) term in this polynomial.

We place the value \(c\) from \(x - c\) to the left of the synthetic division row. For instance, if our divisor is \(x + 3\), \(c = -3\) (since \(x + 3 = x - (-3)\)). We then bring down the first coefficient, multiply it by \(c\), and continue this process, adding vertically each time. The last value you obtain represents the remainder. Synthetic division is a concise and efficient way to determine this remainder, which is also \(f(c)\) according to the Remainder Theorem.

Polynomial Evaluation

Polynomial evaluation is a fundamental process in algebra where you substitute a given value into a polynomial function and calculate the result. This practice helps determine the value of the function for a specific input, and in the context of the Remainder Theorem, it also reveals the remainder when dividing by \(x - c\).

For example, to evaluate \(f(x) = x^4 - 6x^2 + 4x - 8\) at \(c = -3\), we substitute \(-3\) in place of every \(x\) in the expression. Thus, we get:

• \((-3)^4 = 81\),
• \(-6(-3)^2 = -54\),
• \(4(-3) = -12\),
• \(-8 = -8\).

After calculating the value for each term separately, add all these terms together to find the value of the polynomial at \(x = -3\). This method gives you not only \(f(-3)\) but also the remainder in the synthetic division process as per the Remainder Theorem.

Remainder Calculation

Calculating the remainder of a polynomial division using the Remainder Theorem is a straightforward yet powerful process. According to the theorem, the remainder of the division of a polynomial \(f(x)\) by \(x - c\) is simply \(f(c)\). This means you can bypass the entire division operation by directly evaluating \(f\) at \(c\).

Let's understand this with our example: If we need to find the remainder of \(f(x) = x^4 - 6x^2 + 4x - 8\) divided by \(x + 3\), we compute \(f(-3)\). The calculated value, \(f(-3) = 7\), tells us that the remainder is 7. This shortcut saves a lot of time and effort, especially with higher degree polynomials.

In summary, the Remainder Theorem provides a quick way to find the remainder by simple substitution and evaluation. It’s a critical concept that combines synthetic division and polynomial evaluation, emphasizing the importance of evaluating polynomials for practical and theoretical purposes.