Problem 24 - Basic Technical Mathematics with Calculus

Question

Perform the indicated divisions by synthetic division. $$\left(x^{3}-3 x^{2}-x+2\right) \div(x-3)$$

Step-by-Step Solution

Answer

The result is \(x^2 - 1 + \frac{-1}{x-3}\) with a remainder of \(-1\).

Step 1: Set up for Synthetic Division

Identify the divisor \(x - 3\). The root of this expression is \(3\). Note down the coefficients of the dividend polynomial \(x^3 - 3x^2 - x + 2\), which are \(1, -3, -1, 2\). Prepare to perform synthetic division using \(3\) and these coefficients.

Step 2: Perform the Synthetic Division

Write down the number \(3\) and the coefficients \(1, -3, -1, 2\). Draw a horizontal line and place them as such:\[\begin{array}{c|cccc}3 & 1 & -3 & -1 & 2 \\hline & & & & \\end{array}\]Start by bringing down the first coefficient \(1\) below the line. Multiply \(3\) by \(1\) and write the result under the next coefficient \(-3\), then add the numbers in this column.

Step 3: Complete the Synthetic Division

Continue the process:- Bring down \(1\). Add: \(-3 + 3 \times 1 = 0\).- Write \(0\) below the line. Multiply \(3\) by \(0\), which is \(0\), and add it to the next coefficient \(-1\): \(-1 + 0 = -1\).- Write \(-1\) below the line. Multiply \(3\) by \(-1\), which is \(-3\), and add to the last coefficient \(2\): \(2 - 3 = -1\).The synthetic division table now looks like:\[\begin{array}{c|cccc}3 & 1 & 0 & -1 & 2 \\hline & 1 & 0 & -1 & -1 \\end{array}\]

Step 4: Interpret the Results

The bottom row represents the coefficients of the quotient polynomial and the remainder. The coefficients \(1, 0, -1\) correspond to the quotient \(x^2 + 0x - 1\) which simplifies to \(x^2 - 1\).The remainder is \(-1\), therefore, \\[\left(x^3 - 3x^2 - x + 2\right) \div (x - 3) = x^2 - 1 + \frac{-1}{x-3}\]

Step 5: Write the Final Answer

Hence, using synthetic division, the result is:\(x^2 - 1\) with a remainder of \(-1\), expressed as \(x^2 - 1 + \frac{-1}{x-3}\).

Key Concepts

Polynomial Division

Understanding polynomial division is essential when dealing with algebraic expressions.
This method involves dividing one polynomial by another, simplifying it much like how long division works with numbers.

When we talk about polynomial division in algebra, we are generally referring to two methods: long division and synthetic division.

• Long Division: Similar to dividing large numbers. You write down the dividend and the divisor and systematically reduce the dividend by subtracting a multiple of the divisor.
• Synthetic Division: A simplified version of long division, applicable when the divisor is a linear polynomial like \(x - c\). It's faster and more efficient, especially for higher-degree polynomials.

For many students, synthetic division is preferred due to its speed and simplicity, especially when the division involves polynomials of degree three or higher.
In synthetic division, you work only with the coefficients of the polynomials.
It's crucial to list all coefficients, including zero-valued ones, to maintain accuracy.

Remainder Theorem

The remainder theorem is a fundamental concept connected to polynomial division.
According to this theorem, when a polynomial \(f(x)\) is divided by a linear divisor \((x-c)\), the remainder of this division is \(f(c)\).
In simpler terms, you can determine the remainder by simply substituting the root of the divisor into the polynomial instead of completing the division process.

This can save time and confirm your division results. The relationship is expressed as:

• Let a polynomial \(f(x) = x^3 - 3x^2 - x + 2\).
• Divisor is \(x - 3\), thus \(c = 3\).
• By substituting \(3\) into \(f(x)\), you calculate \(f(3)\) to find the remainder.

If you follow through using synthetic division in our exercise, you'll see the remainder is indeed \(-1\), verifying the theorem's utility.

Algebraic Expressions

Algebraic expressions are the foundation of algebra.
They consist of numbers, variables, and operations that come together to form expressions like \(x^3 - 3x^2 - x + 2\).

These expressions often need simplification, like in division problems, to better understand their properties and behavior.

• Terms: Components of an expression separated by plus or minus signs, such as \(x^3\) or \(-3x^2\).
• Coefficients: Numerical factors of each term, e.g., the coefficient of \(-3x^2\) is \(-3\).
• Variables: Symbols representing numbers, usually \(x\) or \(y\) in algebra.

Recognizing these components is key when performing division operations or simplifying expressions.
In synthetic division, understanding which coefficients correspond to each term is crucial for setting up and solving the division accurately.