Problem 38 - College Algebra

Question

Use synthetic division and the Remainder Theorem to find the indicated function value. $$ f(x)=x^{4}-5 x^{3}+5 x^{2}+5 x-6 ; f(2) $$

Step-by-Step Solution

Answer

Based on the synthetic division and Remainder Theorem, the value of the function \(f(2)\) is 0.

Step 1: Setup for Synthetic Division

First we'll set up the synthetic division. We'll use the number 2 (which corresponds to the \(x\) value we're interested in) in the box on the left and all the coefficients of the polynomial on the right, it should be like this:\[\begin{array}{c|cccc}2 & 1 & -5 & 5 & 5 & -6 \end{array}\]

Step 2: Performing Synthetic Division

Start by dropping down the first coefficient \[\begin{array}{c|ccccc|c}2 & 1 & -5 & 5 & 5 & -6 \ & \downarrow & & & & & \end{array}\]Multiply the number in the box (2) by the number you just brought down (1) and write the result underneath the second coefficient of the polynomial; repeat the procedure: add the current number and the number below it (result of multiplication), write the result below and continue to the right until the last column:\[\begin{array}{c|ccccc|r}2 & 1 & -5 & 5 & 5 & -6 \ & \downarrow & 2 & -6 & -2 & 6 & \\hline & 1 & -3 & -1 & 3 & 0 \end{array}\]

Step 3: Using the Remainder Theorem

The Remainder Theorem tells us that the very last number - the remainder after carrying out the synthetic division, is equal to \(f(2)\). Thus, our result is 0.

Key Concepts

Remainder Theorem

The Remainder Theorem is a valuable tool in algebra that links polynomial division with function evaluation. It states that if a polynomial \(f(x)\) is divided by \(x - a\), the remainder of this division is equal to \(f(a)\). This is especially helpful for evaluating the value of a polynomial at a specific point without having to do the evaluation directly.
For example, to find \(f(2)\) for the given polynomial \(f(x)=x^4-5x^3+5x^2+5x-6\), we can use synthetic division with \(x=2\). The remainder from this division will directly give us \(f(2)\). Therefore, in this case, since our remainder is 0, we conclude that \(f(2) = 0\).
This theorem can save a lot of time and effort when dealing with high degree polynomials, making it a powerful shortcut in polynomial arithmetic.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of variables. In the general form, a polynomial function of degree \(n\) can be written as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants called coefficients, and \(x\) is the variable.
These functions are foundational in mathematics because they describe a wide variety of important problems and phenomena. Understanding the behavior of polynomial functions is crucial, as they can range from simple linear functions to complex curves in higher degrees.
In this exercise, \(f(x)=x^4 - 5x^3 + 5x^2 + 5x - 6\) is a fourth-degree polynomial due to its highest power of \(x^4\). Each term contributes differently to the overall shape and properties of the polynomial.

Function Evaluation

Function evaluation is the process of finding the output (or range value) of a function for a given input (or domain value). It involves substituting the input value into the function and calculating the result.
For instance, if you need to evaluate \(f(x)=x^4 - 5x^3 + 5x^2 + 5x - 6\) at \(x=2\), direct calculation would involve substituting 2 into the polynomial:

• Calculate \(2^4 = 16\)
• Subtract \(5 imes 2^3 = 40\)
• Add \(5 imes 2^2 = 20\)
• Add \(5 imes 2 = 10\)
• Subtract 6

The result is 0, confirming what we found using synthetic division.
This direct substitution method, while simple, can be cumbersome for polynomials of higher degrees, which is why concepts like synthetic division are valuable.

Coefficients

Coefficients are the numerical factors in the terms of a polynomial. They dictate how much each power of the variable contributes to the total value of the polynomial function.
In the polynomial \(f(x)=x^4 - 5x^3 + 5x^2 + 5x - 6\), the coefficients are:

• 1 for \(x^4\)
• -5 for \(x^3\)
• 5 for \(x^2\)
• 5 for \(x\)
• -6 (as a constant term)

These coefficients directly affect the shape and zeroes of the polynomial when graphed.
During synthetic division, understanding these coefficients is crucial, as they are the primary elements manipulated. Proper alignment and accurate calculations involving coefficients can lead to correct results, as is shown when finding \(f(2)\) in this exercise.