Problem 32 - College Algebra

Question

Divide using synthetic division. $$ \frac{x^{5}-2 x^{4}-x^{3}+3 x^{2}-x+1}{x-2} $$

Step-by-Step Solution

Answer

The quotient from the synthetic division of \(x^{5}-2 x^{4}-x^{3}+3 x^{2}-x+1\) by \(x-2\) is \(x^{4}+x^{3}+2x^{2}+x+0\), and the remainder is 1. Thus, \(x^{5}-2 x^{4}-x^{3}+3 x^{2}-x+1 = (x-2)(x^{4}+x^{3}+2x^{2}+x+0) + 1\).

Step 1: Setting Up the Synthetic Division

First, write the coefficients of the polynomial \(x^{5}-2 x^{4}-x^{3}+3 x^{2}-x+1\) in a row, which are 1, -2, -1, 3, -1, 1. Then, write down the number \(r\) from the divisor \(x - 2\), which is 2 in this case, outside the row of numbers.

Step 2: Perform the Operations

First, drop down the leading coefficient, which is 1 in this case. Then multiply \(r\) by this number and write the result under the second number in the row (-2). Add these two numbers (-2 and 2) and write the sum in the row below. Repeat this process, each time using the newly calculated number.

Step 3: Construct the Result

The numbers in the last row correspond to the coefficients of the quotient polynomial but decrease the degree. In this case, the quotient polynomial will have degrees from 4 to 0, which correspond to \(x^{4}, x^{3}, x^{2}, x, \) and the constant term.

Key Concepts

Polynomial Long Division

Polynomial long division is a method similar to the traditional arithmetic division but involves polynomials. It is used to divide a polynomial by another polynomial to get a quotient and sometimes a remainder.
Imagine dividing two big numbers using long division. Polynomial long division works somewhat similarly. You want to determine how many times the divisor can "fit" into different segments of the dividend.
Here, the divisor is a polynomial like \(x - 2\). It is important to align terms according to their degree, just like lining up numbers by place value in standard long division.

• Identify: The dividend is the polynomial you want to divide. The divisor is the polynomial you are using to divide.
• Divide: Work from the highest degree of the dividend, finding how many times the leading term of the divisor goes into the leading term of the dividend.
• Subtract: Multiply the entire divisor by this result and subtract from the dividend.
• Repeat: Continue this process with the newly formed polynomial until the degree of the remainder is less than that of the divisor.

This process gives you a quotient and a possible remainder. Synthetic division can be seen as a shorter or simplified version of this traditional method when dividing by linear polynomials.

Coefficients

Coefficients are the numerical factors of the terms in a polynomial. When you see a polynomial such as \(x^5 - 2x^4 - x^3 + 3x^2 - x + 1\), the coefficients are the numbers in front of each term. In this example, these are 1, -2, -1, 3, -1, and 1.
They play a crucial role in polynomial operations like addition, subtraction, multiplication, and especially division. In synthetic division, you focus primarily on these coefficients rather than the full terms.
When setting up synthetic division, you list these coefficients in sequence. For the example polynomial, these will be

• 1 (coefficient of \(x^5\))
• -2 (coefficient of \(x^4\))
• -1 (coefficient of \(x^3\))
• 3 (coefficient of \(x^2\))
• -1 (coefficient of \(x\))
• 1 (constant term)

Manipulating these coefficients correctly during synthetic division allows you to determine the coefficients of the resulting quotient polynomial.

Remainder Theorem

The remainder theorem is a handy concept when working with polynomial division, particularly when a polynomial is divided by a linear divisor of the form \(x - r\).
According to the remainder theorem, the remainder of the division of a polynomial \(f(x)\) by a linear polynomial \(x - r\) is simply \(f(r)\). This means you can find the remainder without fully executing the division.
Using it while performing synthetic division, if our divisor is \(x - 2\), the remainder would be directly linked to evaluating \(f(2)\).
In synthetic division, the final value in the bottom line represents the remainder. If it's zero, the divisor \(x - r\) perfectly divides the polynomial.
This is particularly useful:

• Fast-checking divisibility of polynomials
• Solving polynomial equations by finding roots
• Simplifying complex polynomial divisions with simple calculations

The remainder, from synthetic division, like in our example "1", would tell you how close the quotient polynomial multiplies back to approach the original polynomial.

Division of Polynomials

The division of polynomials can take place through various methods but ultimately yields a quotient and possibly a remainder just like regular arithmetic division does.
We can do this through polynomial long division or the more streamlined synthetic division.

• Through Long Division: Like polynomial long division, where terms are divided, multiplied and subtracted repeatedly until the remainder is of a smaller degree than the divisor.
• Through Synthetic Division: Applied mainly when dividing by linear terms like \(x - c\). It involves using only coefficients, making it faster and easier for specific cases.

The processes themselves involve:

• Determining which part of the polynomial "fits" and subtracting it from the leading terms efficiently.
• Achieving the simplest expression of the polynomial divided by its divisor.
• Helping solve equations, graph polynomials, and analyze polynomial behavior.

Using these methods efficiently allows one to tackle complex algebraic operations in simpler, more manageable steps.