Problem 12 - Precalculus: Functions and Graphs

Question

Use the remainder theorem to find \(f(c)\). $$f(x)=x^{4}+3 x^{2}-12 ; \quad c=-2$$

Step-by-Step Solution

Answer

The remainder is 16.

Step 1: Understand the Remainder Theorem

The remainder theorem states that the remainder when a polynomial \( f(x) \) is divided by \( x - c \) is \( f(c) \). This means we can find \( f(c) \) by substituting \( c \) directly into the polynomial.

Step 2: Substitute c into the Polynomial

Substitute \( c = -2 \) into the polynomial \( f(x) = x^4 + 3x^2 - 12 \). This results in:\[ f(-2) = (-2)^4 + 3(-2)^2 - 12 \]

Step 3: Calculate Each Term Individually

Calculate \((-2)^4\), which equals 16. Calculate \(3(-2)^2\), which equals 12. The constant term is \(-12\).

Step 4: Add the Results Together

Now add the results of each term:\[ 16 + 12 - 12 = 16 \]

Step 5: Conclusion

The value of \( f(c) \), where \( c = -2 \), is 16 according to the remainder theorem.

Key Concepts

Polynomials

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It is a core concept in algebra, allowing us to describe a wide array of functions. For instance, the polynomial depicted in the exercise is:

• \( f(x) = x^4 + 3x^2 - 12 \)

Each term refers to the individual parts of a polynomial, like \( x^4 \), \( 3x^2 \), and \(-12\). The highest power of the variable, known as the degree, dictates the polynomial's behavior. In this case, the degree is 4, derived from the term \( x^4 \). Polynomials can be simple with one term or complex with multiple terms. Understanding polynomials is crucial, as they form the building blocks for many algebraic equations and help in the analysis of functions used in calculus and higher math.
Polynomials are versatile and used in various fields, including physics, engineering, and economics, to model different phenomena. This versatility makes mastering them essential.

Synthetic Substitution

Synthetic substitution is a streamlined method used mainly to evaluate polynomials at a specific value, often following the use of the Remainder Theorem. Unlike traditional substitution, which involves tedious calculations, synthetic substitution reduces complexity by focusing strictly on the coefficients of the polynomial.
To use synthetic substitution, you:

• Write down the coefficients of the polynomial.
• Use the constant from \( x - c \) as your divisor, which simplifies calculations.
• Carry out arithmetic operations using the synthetic division process.

In our example, if implemented, it would make the process of finding \( f(-2) \) from \( f(x) = x^4 + 3x^2 - 12 \) faster. However, in practice for simple calculations like these, direct substitution suffices. Yet, synthetic substitution provides a quicker pathway for larger polynomials and remains a valuable tool in polynomials' arsenal.

Evaluation of Polynomials

Evaluating a polynomial refers to the process of determining its value for a specific variable value, \( c \). Using the Remainder Theorem, this process becomes more straightforward, since the theorem links division of polynomials with substitution.
In our given exercise, you evaluated by:

• Substituting \( c = -2 \) directly into \( f(x) \).
• Calculating each part separately: \((-2)^4\), \(3(-2)^2\), and the constant \(-12\).
• Adding these results together to get the remainder, which is also \( f(-2) \).

This approach simplifies understanding of how varying \( x \) affects the polynomial's value. Evaluations are crucial elsewhere, too—such as in optimizing functions or solving equations in calculus. Using systematic calculations helps streamline the evaluation process, illustrating how changes in inputs directly influence a polynomial's output.