Problem 18 - Algebra for College Students

Question

For Problems \(11-20\), find \(f(c)\) either by using synthetic division and the remainder theorem or by evaluating \(f(c)\) directly. $$ f(n)=-2 n^{5}-9 n^{4}+7 n^{3}+14 n^{2}+19 n-38 \text { and } c=-5 $$

Step-by-Step Solution

Answer

Using synthetic division, \(f(-5) = -33\).

Step 1: Understand the Problem

We are given a polynomial function \(f(n) = -2n^5 - 9n^4 + 7n^3 + 14n^2 + 19n - 38\). We need to find \(f(-5)\) using either the remainder theorem through synthetic division or by direct evaluation.

Step 2: Set Up for Synthetic Division

Using synthetic division, the divisor is \(n - c = n + 5\). List the coefficients of the polynomial: \(-2, -9, 7, 14, 19, -38\). Use \(-5\) as the synthetic divisor.

Step 3: Perform Synthetic Division

Bring down the first coefficient, \(-2\). Multiply \(-2\) by \(-5\), which gives \(10\). Add this to the next coefficient, \(-9\), to get \(1\). Continue this process:1. Bring down \(-2\).2. Multiply and add: \((-5) \times (-2) = 10,\) \(-9 + 10 = 1\).3. \((-5) \times (1) = -5,\) \(7 + (-5) = 2\).4. \((-5) \times (2) = -10,\) \(14 + (-10) = 4\).5. \((-5) \times (4) = -20,\) \(19 + (-20) = -1\).6. \((-5) \times (-1) = 5,\) \(-38 + 5 = -33\).The remainder is \(-33\).

Step 4: Conclusion using Remainder Theorem

The remainder theorem states that the remainder of the division of a polynomial \(f(n)\) by \(n - c\) is \(f(c)\). Thus, \(f(-5) = -33\).

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing polynomials, particularly useful when dividing by a linear factor of the form \(n - c\). Unlike long division, synthetic division reduces the division process to a simple sequence of arithmetic operations, involving only the coefficients of the polynomial. Here's how you can understand each step of synthetic division:

• First, identify your divisor as \(n - c\). For example, if you are dividing by \(n + 5\), your divisor is \(n - (-5)\).
• Write down the coefficients of the polynomial. For instance, with its function \(-2n^5 -9n^4 + 7n^3 + 14n^2 + 19n - 38\), the coefficients would be \([-2, -9, 7, 14, 19, -38]\).
• Use the value \(c\) (which is \(-5\) in this case) as the synthetic divisor and start the process of sequential multiplying and adding. Begin by bringing down the leading coefficient to the bottom row.

Synthetic division is straightforward and efficient, transforming what can be complex algebraic manipulation into simple arithmetic.

Remainder Theorem

The Remainder Theorem is a valuable shortcut to determine the remainder of a polynomial division without executing a complete division. It states that the remainder of the division of a polynomial \(f(n)\) by \(n - c\) is simply \(f(c)\).This theorem takes advantage of the fact that when a polynomial \(f(n)\) is divided by \(n - c\), any remainder is precisely the value of \(f(c)\). In practical terms:

• Once you perform synthetic division, the remainder that you obtain is directly \(f(c)\).
• In our case, after carrying out synthetic division on the polynomial with \(-5\), we find that the remainder is \(-33\). Thus, by the Remainder Theorem, \(f(-5) = -33\).

This method is not only quick but also quite insightful, as it links the analytical process of division with the direct evaluation of polynomial functions.

Polynomial Function

A polynomial function is an expression featuring a sum of terms, where each term includes a variable raised to a non-negative integer power and multiplied by a coefficient. Polynomials are the building blocks of algebra, and they come in many forms, such as quadratic, cubic, quartic, etc., depending on the degree, which is the highest power of the variable in the expression.In the polynomial function given:\(-2n^5 - 9n^4 + 7n^3 + 14n^2 + 19n - 38,\)

• The highest power of the polynomial is 5, making it a quintic polynomial.
• Each term contributes to the shape of the graph of the polynomial, affecting its zeroes, turning points, and end behavior.
• The coefficients (e.g., \(-2, -9, 7,\) etc.) control the steepness and direction of each segment of the polynomial.

Understanding polynomial functions is essential for solving equations and inequalities and for analyzing mathematical models of real-world phenomena.

Direct Evaluation

Direct evaluation of a polynomial at a specific value involves substituting the value into the polynomial and calculating the result directly. This method can be swift and straightforward, especially when the degree of the polynomial is not too high.When given a polynomial function, for example, \(f(n) = -2n^5 - 9n^4 + 7n^3 + 14n^2 + 19n - 38\), and a number to evaluate such as \(c = -5\), follow these steps:

• Substitute \(-5\) into the polynomial wherever you see \(n\).
• Calculate each term separately:
  • For the term \(-2n^5\), compute \(-2(-5)^5\).
  • Repeat similarly for each term, i.e., \(-9(-5)^4, 7(-5)^3,\) etc.
• Combine all the results to find \(f(-5)\).

The calculated result should match the remainder found via synthetic division. Direct evaluation gives exactly the same numerical outcome as the remainder theorem, but emphasizes understanding of the components of the polynomial.