Problem 16 - 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. $$ f(n)=-3 n^{6}-2 \text { and } c=-3 $$

Step-by-Step Solution

Answer

Evaluate directly: \(f(-3) = -3(-3)^6 - 2 = -2185\.

Step 1: State the Function and the Given Value

We are given the polynomial function \(f(n) = -3n^6 - 2\) and need to find \(f(c)\) where \(c = -3\).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

Evaluate directly: \(f(-3) = -3(-3)^6 - 2 = -2185\.

Key Concepts

Synthetic Division

Synthetic division is a shorthand method of polynomial division, typically used when dividing by a linear factor. It simplifies complex long division and is especially useful for evaluating polynomial functions and finding zeros.To perform synthetic division:- Write down the coefficients of the polynomial.- Substitute the value you are evaluating in place of the variable. For example, if evaluating with a given value of \(c = -3\), you substitute this into the synthetic division setup.- Carry through the synthetic division process where you repeatedly multiply and add down the columns.While this process is efficient, synthetic division specifically only works well when dividing a polynomial by another polynomial of degree 1. It's a powerful technique in polynomial algebra.

Remainder Theorem

The Remainder Theorem provides a quick way to find the remainder of a polynomial \(f(x)\) divided by the linear polynomial \(x - c\). The theorem states that when a polynomial \(f(x)\) is divided by \(x - c\), the remainder of this division is \(f(c)\).This means that:- Evaluating \(f(c)\) for your polynomial can be achieved without performing full polynomial division.- You simply substitute \(c\) directly into the polynomial, which quickly provides the remainder.In the context of synthetic division, the final number at the bottom of the setup is the remainder when the polynomial is divided by \(x - c\). For the problem \(f(n)\), the Remainder Theorem validates the use of direct substitution when \(x = c\).

Direct Evaluation

Direct evaluation involves substituting the value directly into the polynomial and computing the result. It bypasses the techniques of synthetic division or other forms of manipulation. This method is straightforward:- Substitute \(-3\) in place of \(n\) for the given polynomial \(-3n^6 - 2\).- Calculate directly to find \((-3)(-3)^6 - 2\) which leads to the direct answer for \(f(c)\).Although this requires manual computation, it guarantees that the result represents the exact value of the polynomial at that point. Direct evaluation is elemental in understanding how polynomials behave at specific values.

Polynomial Degree

The degree of a polynomial refers to the highest power of the variable in the expression. It essentially tells us about the polynomial's structure and its general behavior.For example, the degree of \(-3n^6 - 2\) is 6 due to the \(-3n^6\) term:- The degree is 6, which is important because it influences the number of turning points and the general shape of the polynomial graph.- The polynomial's behavior as \(n\) approaches infinity or negative infinity is largely dictated by the term with the highest degree.Understanding the polynomial degree is crucial for deep understanding of polynomial functions and solving related algebraic problems. This knowledge aids in anticipating the behavior of functions when evaluated directly or through division methods.