Problem 102 - Algebra for College Students

Question

Discussion. Use synthetic division to find the quotient when \(x^{5}-1\) is divided by \(x-1\) and the quotient when \(x^{6}-1\) is divided by \(x-1 .\) Observe the pattern in the first two quotients and then write the quotient for \(x^{9}-1\) divided by \(x-1\) without dividing.

Step-by-Step Solution

Answer

The quotient for \(x^9 - 1\) divided by \(x-1\) is \ x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 \.

Step 1: Set Up Synthetic Division for First Problem

To divide \(x^{5}-1\) by \(x-1\), we use synthetic division. Here, the coefficient of the divisor \(x-1\) is 1. The coefficients of the polynomial \(x^{5}-1\) are \[1, 0, 0, 0, 0, -1\].

Step 2: Perform Synthetic Division for First Problem

Set up the synthetic division as follows: Write down the coefficients \[1, 0, 0, 0, 0, -1\]. Bring down the leading coefficient (1) to the bottom row. Multiply this number by the root (1) and write the result under the next coefficient. Add the numbers in the column. Repeat this process for each coefficient.\( \begin{array{r|rrrrrr} 1 & 1 & 0 & 0 & 0 & 0 & -1 \ \hline & 1 & 1 & 1 & 1 & 1 & 0 \end{array} \)\.The quotient is \x^4 + x^3 + x^2 + x + 1\.

Step 3: Set Up Synthetic Division for Second Problem

To divide \(x^{6}-1\) by \(x-1\), use synthetic division again. The coefficients of \(x^{6}-1\) are \[1, 0, 0, 0, 0, 0, -1\].

Step 4: Perform Synthetic Division for Second Problem

Set up the synthetic division as follows: Write the coefficients \[1, 0, 0, 0, 0, 0, -1\]. Bring down the leading coefficient (1) to the bottom row. Multiply this number by the root (1) and write the result under the next coefficient. Add the numbers in the column. Repeat for each coefficient.\( \begin{array{r|rrrrrrr} 1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 \ \hline & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \end{array} \)\.The quotient is \x^5 + x^4 + x^3 + x^2 + x + 1\.

Step 5: Observe the Pattern

Based on the first two problems, the quotient when dividing \(x^n - 1\) by \(x-1\) appears to be \(x^{n-1} + x^{n-2} + \ldots + x + 1\).

Step 6: Write the Quotient for Third Problem

Without performing synthetic division, the quotient when \(x^9 - 1\) is divided by \(x-1\) is based on the observed pattern \ x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 \.

Key Concepts

polynomials

A polynomial is an expression that consists of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication.
The general form is a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 . Here:

• a_n, a_{n-1}, ..., a_0 are coefficients.
• x is the variable.
• n is a non-negative integer known as the degree of the polynomial.

Polynomials can have one or multiple variables. High school algebra often focuses on single-variable polynomials.

division of polynomials

Dividing polynomials involves finding how many times a polynomial (the divisor) can be subtracted from another polynomial (the dividend).
One effective method for this is synthetic division, especially when the divisor is in the form of x-c.
Steps to perform synthetic division:

• Write down the coefficients of the dividend polynomial.
• Identify the constant term 'c' from the divisor.
• Perform iterative steps to divide and simplify, as demonstrated in the problem solution.

In synthetic division, we reduce the problem to dealing with only coefficients and systematically find the quotient.

algebraic patterns

Identifying algebraic patterns simplifies solving polynomial problems.
In the problems provided, a pattern emerges: dividing x^n - 1 by x -1 results in a quotient x^{n-1} + x^{n-2} + ... + x + 1 .
This pattern allows us to predict results without extensive calculations.
Recognizing these patterns reduces your work and helps in quickly solving similar polynomial problems.

quotients of polynomials

The quotient is the result obtained when one polynomial is divided by another.
In the synthetic division examples given:

• x^5 - 1 divided by x - 1 results in x^4 + x^3 + x^2 + x + 1
• x^6 - 1 divided by x - 1 results in x^5 + x^4 + x^3 + x^2 + x + 1

These demonstrate that the quotient of x^n -1 divided by x-1 always takes the form, x^{n-1} + x^{n-2} + ... + x + 1 .
Observing such patterns in quotients makes solving polynomial divisions easier and more predictable.