Problem 41 - Precalculus : Building Concepts and Connections

Question

Find the remainder when \(x^{7}+7\) is divided by \(x-1.\)

Step-by-Step Solution

Answer

The remainder when \(x^{7}+7\) is divided by \(x-1\) is 8.

Step 1: Identify application of the Polynomial Remainder Theorem

The task has asked for the remainder when \(x^{7}+7\) is divided by \(x-1\). According to Polynomial Remainder Theorem, to get the remainder we just have to substitute the value of x in the given polynomial which makes the divisor equal to zero.

Step 2: Implement Polynomial Remainder Theorem

So, let's substitute \(x = 1\) in the polynomial \(x^{7}+7\) to get the remainder.

Step 3: Final Calculation

Substituting \(x = 1\) in \(x^{7}+7\), we get \(1^{7} + 7 = 1 + 7 = 8\).

Key Concepts

Polynomial Division

Polynomial division is a process used to divide two polynomials, much like traditional long division with numbers. This technique is vital in algebra, where it is frequently used to simplify expressions, solve polynomial equations, or find the remainder when one polynomial is divided by another.

Consider you have a polynomial, denoted as the dividend, and you want to divide it by another polynomial, called the divisor. The goal is to determine how many times the divisor can go into the dividend, which results in the quotient, and what is left over, known as the remainder.

• The dividend is the polynomial being divided.
• The divisor is the polynomial by which you are dividing.
• The quotient is the result of the division.
• The remainder is what's left when the division process is complete.

Remainder Calculation

The process of calculating the remainder in polynomial division can be greatly simplified by the Polynomial Remainder Theorem. This theorem outlines a shortcut where, instead of performing long polynomial division, you can plug in a specific value of 'x' into the dividend polynomial to find the remainder directly.

To apply the theorem effectively:

• First, identify the zero of the divisor polynomial. This is done by setting the divisor equal to zero and solving for 'x'.
• Then, substitute this 'x' value into the dividend polynomial.
• The result of the substitution gives you the remainder of the division.

This method is markedly faster than long polynomial division, especially with higher-degree polynomials, and it only applies when dividing by a linear divisor of the form 'x - c'. Using this theorem, complicated calculations can be streamlined, saving time and reducing potential errors.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables, multiplied by coefficients. These functions are an essential concept in algebra and calculus due to their versatility and the ease with which they can be manipulated and graphed.

These functions have various components:

• The degree of the polynomial is the highest power of the variable in the expression.
• Coefficients are the numbers multiplying the variables or powers of variables.
• Terms are the individual parts of the polynomial, separated by addition or subtraction.

Polynomial functions often describe real-world phenomena such as motion, and they can be used in a wide array of applications ranging from physics to economics. Understanding how to manipulate these functions, including operating division, and calculating remainders simplifies the study of more advanced mathematical concepts.