Problem 37 - College Algebra

Question

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

Step-by-Step Solution

Answer

Hence, applying synthetic division and using the Remainder Theorem, we find that \(f(3)\) is equal to the remainder after the synthetic division.

Step 1: Prepare the synthetic division table

Arrange the coefficients of the polynomial \(f(x)\) in descending order based on the power of \(x\) as the first row of a table. For the polynomial \(f(x) = x^{4} + 5x^{3} + 5x^{2} - 5x - 6\) the coefficients are [1, 5, 5, -5, -6]. On the left of the table, write down \(x = 3\) which is the value we want to evaluate.

Step 2: Carry out the synthetic division

Bring down the leading coefficient of 1 to the bottom row. Multiply 3 (value of x) by the value just carried down (1), and write the result under the second coefficient in the top row. Add the number you wrote under the second coefficient to the second coefficient itself, and bring the sum to the bottom row. Continue these steps until you've worked through to the final column.

Step 3: Interpret the result

The last value on the bottom row (the remainder of the synthetic division) represents the value of the function at \(x = 3\). According to the Remainder Theorem, if a polynomial \(f(x)\) is divided by \(x - a\), then the remainder is \(f(a)\). In this case, \(a = 3\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a concept in algebra that connects the outcomes of polynomial division to function evaluation. When you divide a polynomial function, such as \(f(x)=x^4+5x^3+5x^2-5x-6\), by a linear factor of the form \(x - a\), the remainder gives you the value of the polynomial function for \(f(a)\). Simply put, if you want to find \(f(3)\), you can perform polynomial division of \(f(x)\) by \(x-3\), and the remainder of this division is the result of \(f(3)\).

This is particularly useful because it allows for an alternative to direct substitution, which might be more complex for higher power polynomials. Instead, synthetic division, a more efficient process, can be used to determine the remainder and therefore the value of the function at \(x=a\), as demonstrated in the original exercise.

Polynomials

Polynomials are mathematical expressions that consist of variables (often referred to as indeterminates) and coefficients, that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial is the function given in the original exercise, \(f(x)=x^4+5x^3+5x^2-5x-6\), where the highest exponent of the variable \(x\) determines the degree of the polynomial—in this case, a fourth degree polynomial.

Understanding the structure of polynomials is essential when employing synthetic division. A polynomial is written in standard form when its terms are ordered from highest to lowest degree of \(x\). The coefficients of these terms, in this order, are the starting point for synthetic division.

Function Evaluation

Function evaluation is the process of finding the output value of a function for a particular input value. For the polynomial function in question, \(f(x)\), the function evaluation at \(x = 3\) requires substituting 3 for all instances of \(x\) in the polynomial and simplifying. However, direct substitution can sometimes be tedious and prone to error, especially with higher degree polynomials.

As an alternative, synthetic division streamlines this process by providing a systematic way to calculate the remainder, which equates to the function's value at a given input when used in conjunction with the Remainder Theorem. By using the synthetic division as outlined in the exercise, we can quickly and accurately determine \(f(3)\) without the need for the potentially more complex direct computation.