Problem 49 - College Algebra with Modeling and Visualization

Question

Use the remainder theorem to find the remainder when \(f(x)\) is divided by the given \(x-k\) $$f(x)=4 x^{3}-x^{2}+4 x+2 \quad x+2$$

Step-by-Step Solution

Answer

The remainder is -42.

Step 1: Identify 'k'

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - k \), the remainder of that division is \( f(k) \). Here, we are given \( x + 2 \). Rewrite \( x + 2 \) as \( x - (-2) \), where \( k = -2 \).

Step 2: Substitute 'k' in f(x)

Substitute \( k = -2 \) into the polynomial \( f(x) = 4x^3 - x^2 + 4x + 2 \) to find the remainder. \[ f(-2) = 4(-2)^3 - (-2)^2 + 4(-2) + 2 \].

Step 3: Simplify the Expression

Calculate \( f(-2) \) step by step: - Calculate \((-2)^3 = -8\). Therefore, \(4(-2)^3 = 4(-8) = -32\). - Calculate \((-2)^2 = 4\). Therefore, \(-(-2)^2 = -4\). - Calculate \(4(-2) = -8\). - Finally, add: \(-32 - 4 - 8 + 2\).

Step 4: Calculate the Final Remainder

Combine all results from the expression: \(-32 - 4 - 8 + 2 = -42\). Thus, the remainder when \( f(x) \) is divided by \( x + 2 \) is \(-42\).

Key Concepts

Polynomial Division

Polynomial division can be likened to long division, but instead of numbers, we are working with algebraic expressions. When dividing one polynomial by another, we aim to find both the quotient and the remainder. The general form of dividing a polynomial \( f(x) \) by a linear polynomial \( x-k \) is crucial here.

• The polynomial \( f(x) \) stands for the dividend in this setup.
• The expression \( x-k \), or equivalently \( x-c \), represents the divisor.
• The result of this division will yield a quotient polynomial and potentially a non-zero remainder.

Understanding polynomial division is fundamental as it lays the groundwork for concepts like the Remainder Theorem, enhancing our problem-solving skills in algebra.

Remainder Calculation

The remainder calculation is made straightforward thanks to the Remainder Theorem. This theorem simplifies the process of finding the remainder of a polynomial division, especially when dealing with linear divisors, i.e., expressions in the form \( x-k \).

• According to the Remainder Theorem, for a polynomial \( f(x) \) divided by \( x - k \), the remainder is simply \( f(k) \).
• This provides a direct method to find the remainder without performing full polynomial division.
• In our exercise, the task was to divide \( f(x) = 4x^{3} - x^{2} + 4x + 2 \) by \( x + 2 \).

By treating \( x + 2 \) as \( x - (-2) \), we deduced that the remainder is \( f(-2) \). This allows us to bypass tedious calculations and focus directly on substituting and evaluating the expression.

Substitution Method

The substitution method involves calculating the value of the polynomial at a specific point, which here corresponds to evaluating \( f(k) \) as prescribed by the Remainder Theorem.

• Begin by identifying the correct value of \( k \) from the divisor \( x - k \).
• In our example, \( k = -2 \), derived from \( x + 2 \).
• Substitute \( k = -2 \) into the polynomial function: \( f(x) = 4x^{3} - x^{2} + 4x + 2 \).
• Step-by-step simplification involves performing arithmetic operations sequentially to ensure accuracy.

Through substitution and simple arithmetic, we calculated \( f(-2) = -42 \). This value represents the remainder when dividing the polynomial by \( x+2 \). Such methods streamline finding quick and dependable results in algebra, especially useful in tasks involving higher-degree polynomials.