Problem 50 - College Algebra with Modeling and Visualization

Question

Use the remainder theorem to find the remainder when \(f(x)\) is divided by the given \(x-k\) $$f(x)=-x^{4}+4 x^{3}-x+3 \quad x-3$$

Step-by-Step Solution

Answer

The remainder is 27.

Step 1: Understand the Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x-k \), the remainder of this division is given by \( f(k) \). Thus, to find the remainder when \( f(x) \) is divided by \( x-3 \), we must evaluate \( f(3) \).

Step 2: Substitute \( x = 3 \) into the Polynomial

Plug \( x = 3 \) into the polynomial \( f(x) = -x^4 + 4x^3 - x + 3 \). This results in:\[f(3) = -(3)^4 + 4(3)^3 - 3 + 3\]

Step 3: Calculate Each Term

Calculate the terms separately:\(-x^4 = -(3)^4 = -81\)\(4x^3 = 4(3)^3 = 4 \times 27 = 108\)The linear term is \(-3\) and the constant term is \(3\).

Step 4: Combine the Results

Combine the results of each term to find \( f(3) \):\[f(3) = -81 + 108 - 3 + 3\]Add these together to simplify:\[= 27 \]

Step 5: Final Answer

Thus, the remainder when \( f(x) = -x^4 + 4x^3 - x + 3 \) is divided by \( x-3 \) is \(27\).

Key Concepts

Polynomial Division

Polynomial division is an essential concept in algebra, much like long division is in arithmetic. When we are given a polynomial, such as \(f(x) = -x^4+4x^3-x+3\), and asked to divide it by another polynomial (often in the form \(x - k\)), we're essentially looking for two things: the quotient and the remainder. The polynomial we are dividing by, in this exercise \(x-3\), is often referred to as the divisor, while the polynomial we are dividing, \(f(x)\), is the dividend.
The goal is to express the original polynomial as \(f(x) = (x-k)Q(x) + R\), where \(Q(x)\) is the quotient, and \(R\) is the remainder. One practical method to perform polynomial division is synthetic division, especially when the divisor is in the form \(x-k\). This method simplifies calculations significantly and is often quicker than traditional long division.

It's important to note that the degree of the remainder (R) is always less than the degree of the divisor (here, a linear polynomial). In this exercise, since we are dividing by \(x-3\), the remainder is simply a constant value, and the process of finding this involves using tools like the Remainder Theorem.

Remainder Calculation

The Remainder Theorem provides a swift approach for finding remainders in polynomial division without fully dividing the polynomials. It asserts that the remainder of dividing \(f(x)\) by \(x-k\) is precisely \(f(k)\). This theorem stems from the property of polynomials: if you substitute \(k\) into the polynomial, the outcome equals the remainder when divided by \(x-k\).
For example, in our exercise, to find the remainder when \(-x^4+4x^3-x+3\) is divided by \(x-3\), we substitute 3 for \(x\) in the polynomial. This technique avoids the lengthy process of division and gives the result instantly.

After substitution, the calculation involves simply evaluating the polynomial at this point, as shown:

• Compute \(-x^4\) as \(-(3)^4 = -81\)
• Calculate \(4x^3\) as \(4(3)^3 = 108\)
• Combine the remaining linear and constant terms \(-3\) and \(+3\)

The final sum of these calculated values gives the remainder, ensuring the simplicity and elegance of the Remainder Theorem.

Evaluating Polynomials

Evaluating a polynomial entails substituting a particular value into the polynomial and computing the result. This process is straightforward yet critical, particularly when applying the Remainder Theorem. The primary step involves replacing the variable \(x\) with the given number, computing the powers of \(x\), multiplying by the coefficients, and finally summing all terms.
In practice, let’s follow the exercise where \(f(x) = -x^4+4x^3-x+3\) needs to be evaluated at \(x = 3\). The steps are:

• Substitute 3 in place of \(x\).
• Evaluate each term:
  • For \(-x^4\), calculate \(-81\).
  • For \(4x^3\), calculate \(108\).
  • The terms \(-x\) and \(+3\) become \(-3\) and \(+3\) respectively.
• Add these results: \(-81 + 108 - 3 + 3 = 27\).

The result is the value of the polynomial at \(x = 3\), which in the context of the Remainder Theorem, is the same as the remainder of the division. Understanding polynomial evaluation is key to mastering polynomial division and applying the Remainder Theorem effectively.