Problem 10 - Precalculus: Functions and Graphs

Question

Use the remainder theorem to find \(f(c)\). $$f(x)=2 x^{3}+4 x^{2}-3 x-1 ; \quad c=3$$

Step-by-Step Solution

Answer

The value of \(f(3)\) is 80.

Step 1: State the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \(f(x)\) by \(x-c\) is \(f(c)\). Thus, finding \(f(c)\) will give us the remainder of the polynomial when divided by \(x-c\).

Step 2: Calculate \(f(3)\)

Substitute \(c = 3\) into the polynomial \(f(x) = 2x^3 + 4x^2 - 3x - 1\). So, calculate: \[f(3) = 2(3)^3 + 4(3)^2 - 3(3) - 1\]

Step 3: Simplification of \(f(3)\)

First, calculate \((3)^3 = 27\) and \((3)^2 = 9\). Substitute these values back into the equation: \[f(3) = 2 \times 27 + 4 \times 9 - 3 \times 3 - 1\]

Step 4: Complete the Calculation

Calculate each term: - \(2 \times 27 = 54\), - \(4 \times 9 = 36\), - \(3 \times 3 = 9\). Now, plug these into the equation to get: \[f(3) = 54 + 36 - 9 - 1\]

Step 5: Find the Result of \(f(3)\)

Add and subtract the numbers: \[f(3) = 54 + 36 - 9 - 1 = 80\].

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, similar to the way we perform division with numbers. However, dividing polynomials involves more steps and understanding. The main goal is to express a polynomial as the product of a divisor and a quotient, plus a remainder, which mirrors the structure: \[ \frac{\text{Dividend} - \text{Remainder}}{\text{Divisor}} = \text{Quotient} \]. When dividing a polynomial, the remainder has a smaller degree than the divisor. This is a crucial point, as it is what makes the Remainder Theorem applicable.

• Polynomial division can be done using long division or synthetic division techniques.
• The degree of the remainder is always less than the degree of the divisor.
• The division process helps to simplify polynomials in equations and calculus.

Understanding polynomial division can make comprehending more complex concepts like the Remainder Theorem much easier. Every step in polynomial division counts towards finding the quotient and remainder.

Evaluating Polynomials

Evaluating polynomials involves finding the value of the polynomial function at a particular point, such as at a specific value of x. This process is incredibly important, not only in finding solutions to equations but also in applying mathematical models to real-world situations. For example, if we have a polynomial function \(f(x)\), and we need to find \(f(c)\), we substitute \(x\) with \(c\) in the polynomial. The idea is to compute:

• First replace every instance of \(x\) with \(c\).
• Carry out arithmetic operations in the polynomial after substitution.

In our exercise, evaluating \( f(x) = 2x^3 + 4x^2 - 3x - 1 \) at \( c = 3 \), required substituting \( 3 \) and simplifying. This process of evaluating polynomials provides not only a result but also insight into polynomial behavior at specific points.

Polynomial Remainder

The polynomial remainder is what remains after performing polynomial division. According to the Remainder Theorem, the remainder obtained when a polynomial \(f(x)\) is divided by a linear divisor \(x-c\) is simply \(f(c)\). This direct computation of the remainder without actual division saves time and effort.When considering the Remainder Theorem:

• The theorem only applies when dividing by a linear factor such as \(x-c\).
• The process involves evaluating the polynomial at \(c\) to find the remainder.
• This remainder helps determine how closely a point \((c, f(c))\) fits the polynomial function.

The theorem illustrates an elegant connection between polynomial evaluation and division, showing that even complex algebraic expressions can be simplified.

Polynomial Functions

Polynomial functions are crucial building blocks in mathematics, represented by algebraic expressions involving a sum of powers of variables. Each term in a polynomial function is composed of a coefficient and a variable raised to a non-negative integer power. Polynomials can be classified by degree, with the highest power determining it. For instance, a polynomial like \(f(x) = 2x^3 + 4x^2 - 3x - 1\) is a cubic polynomial with four terms. Here’s why polynomial functions are important:

• They appear in a wide range of scientific calculations and mathematical problems.
• Understanding them leads to insights in calculus, algebra, and even physics.
• They model real-life situations, like physics equations or economic models.

Polynomial functions not only form the backbone of algebra but also help in deriving important mathematical theorems and facilitating computational techniques.