Problem 11 - Basic Technical Mathematics with Calculus

Question

Find the remainder using the remainder theorem. Do not use synthetic division. $$\left(R^{4}+R^{3}-9 R^{2}+3\right) \div(R-3)$$

Step-by-Step Solution

Answer

The remainder is 30.

Step 1: Understanding the Remainder Theorem

The remainder theorem states that if a polynomial \( f(x) \) is divided by \( x - c \), the remainder of this division is \( f(c) \). In this exercise, we aim to find the remainder when \( R^4 + R^3 - 9R^2 + 3 \) is divided by \( R - 3 \). According to the theorem, this remainder will be equal to \( f(3) \).

Step 2: Substitute \( R = 3 \) into the Polynomial

To find \( f(3) \), substitute \( R = 3 \) into the given polynomial: \[ f(3) = 3^4 + 3^3 - 9(3)^2 + 3 \]

Step 3: Calculate Each Term

Now compute each term: 1. \( 3^4 = 81 \)2. \( 3^3 = 27 \)3. \( 9(3)^2 = 81 \)4. Constant term = 3.

Step 4: Add the Terms Together

Substitute the calculated values back into the expression to find \( f(3) \):\[ f(3) = 81 + 27 - 81 + 3 \] Start by adding the terms: \( 81 + 27 = 108 \). Then, subtract \( 81 \): \( 108 - 81 = 27 \). Finally, add \( 3 \): \( 27 + 3 = 30 \).

Step 5: Interpreting the Result

The calculation shows that when the polynomial is evaluated at \( R = 3 \), the result is 30. According to the remainder theorem, this value is the remainder when \( R^4 + R^3 - 9R^2 + 3 \) is divided by \( R - 3 \).

Key Concepts

Polynomial Division

Polynomial division is a method used in mathematics to divide one polynomial by another. It's akin to the long division process used in arithmetic. The goal is to divide a polynomial, called the dividend, by another polynomial, termed the divisor. Typically, the division process results in a quotient and possibly a remainder. The Remainder Theorem simplifies this process significantly by using a straightforward calculation to find the remainder without executing the full division.

The Remainder Theorem states that the remainder of the division of a polynomial \( f(x) \) by \( x - c \) can be calculated simply as \( f(c) \). Instead of dividing, we substitute the value into the polynomial to determine the remainder, which makes the process much more efficient.

Algebra

Algebra is an essential branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols represent numbers, and the way they are manipulated follows specific algebraic operations like addition, subtraction, multiplication, and division.

In the context of polynomial division, algebra provides the theoretical groundwork that underpins methods like the Remainder Theorem. Understanding how these symbols and expressions work allows us to simplify complex polynomial functions, solve equations, and find specific values for variables.

With algebra, we can express the process of dividing polynomials in a standardized way, enabling problem-solving across various mathematical fields.

Evaluating Polynomials

Evaluating polynomials involves finding the value of a polynomial function at a given point. This is achieved by substituting the chosen value into the polynomial and carrying out the necessary calculations.

To evaluate polynomials effectively, one must understand both the coefficients and the variable substitutions. This process is integral to using the Remainder Theorem, which provides a direct way to find remainders by substitution.

In our example, evaluating \( f(R) \) at \( R = 3 \) involves substituting 3 into the polynomial \( R^4 + R^3 - 9R^2 + 3 \) and calculating the resultant value. This substitution step replaces the complex division operation and offers a simpler path to the desired result. Each substituted expression (such as \( 3^4 \) and \( 9 \times 3^2 \)) is computed individually before being combined to find the final outcome.