Problem 53 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to find the quotient and remainder. $$ \frac{x^{4}-22}{x+2} $$

Step-by-Step Solution

Answer

Quotient: \(x^3 - 2x^2 + 4x - 8\); Remainder: \(-6\).

Step 1: Set up Synthetic Division

Write the coefficients of the polynomial \(x^4 - 22\) in descending order: \([1, 0, 0, 0, -22]\). Since there is no \(x^3\), \(x^2\), or \(x\) term, their coefficients are zeros. The divisor is \(x + 2\), which means we use \(-2\) in synthetic division.

Step 2: Perform Synthetic Division

We start by bringing down the leading coefficient, 1, to the bottom row. Multiply \(-2\) with 1, and write the result under the next coefficient (0). Add the numbers in the column, and repeat this process for each column. The sequence will be: 1. Bring down 1.2. Multiply \(-2\) with 1, get -2, and add to 0 (next coefficient), result is -2.3. Multiply \(-2\) with -2, get 4, add to 0, result is 4.4. Multiply \(-2\) with 4, get -8, add to 0, result is -8.5. Multiply \(-2\) with -8, get 16, add to -22, result is -6.

Step 3: Interpret the Results

The bottom row of numbers, \([1, -2, 4, -8, -6]\), represents the coefficients of the quotient and the remainder. The quotient is \(x^3 - 2x^2 + 4x - 8\). The remainder is \(-6\).

Step 4: Write the Final Answer

The synthetic division yields a quotient of \(x^3 - 2x^2 + 4x - 8\) and a remainder of \(-6\). This means:\[\frac{x^4 - 22}{x+2} = x^3 - 2x^2 + 4x - 8 - \frac{6}{x+2}\]

Key Concepts

Polynomial Division

Polynomial division resembles the division of numbers but applies it to polynomials. In doing so, just like you'd divide numbers in arithmetic, you must first determine how many times the divisor can go into the dividend.

When using polynomial division, it can be lengthy and cumbersome to perform manually. This is where synthetic division comes in to make the process faster and easier. For dividing a polynomial like \(x^4 - 22\) by \(x + 2\), synthetic division simplifies the task with a clear process.

• List the coefficients: For \(x^4 - 22\), the coefficients are \([1, 0, 0, 0, -22]\). Notice we include zeros for missing terms like \(x^3\), \(x^2\), and \(x\).
• Select the value of \(r\): Since the divisor is \(x + 2\), use \(-2\) for synthetic division because \(r\) is the zero of the divisor.

This method streamlines the division, focusing only on coefficients and combining results through simple arithmetic. It yields a quotient and a remainder, succinctly encapsulating the division's outcome.

Quotient and Remainder

In polynomial division, understanding the results means interpreting both the quotient and the remainder effectively. The quotient represents the main result of division, while the remainder reveals what's left over.

When our exercise divides \(x^4 - 22\) by \(x + 2\), synthetic division provides two key outputs:

• Quotient: This is the polynomial form derived from the coefficients resulting from our process. The bottom row from synthetic division, \([1, -2, 4, -8]\), translates to a quotient polynomial of \(x^3 - 2x^2 + 4x - 8\).
• Remainder: This is any leftover value not explained by the quotient portion, given here as \(-6\). When a remainder is present, the division isn't exact.

To express a polynomial division with a remainder, use the relation: \[\frac{Dividend}{Divisor} = Quotient + \frac{Remainder}{Divisor}\]It concisely shows how every piece fits together, offering a complete view of the division's outcome.

Rational Expressions

Rational expressions involve fractions or ratios of polynomials. They operate similarly to numerical fractions, encompassing concepts of numerators and denominators.

The division we performed creates a rational expression when the remainder isn’t zero. Consider the result of our previous division task:

• Main polynomial quotient: \(x^3 - 2x^2 + 4x - 8\)
• Remainder fraction: \(-\frac{6}{x+2}\)

Combining these, the full expression is:\[\frac{x^4 - 22}{x+2} = x^3 - 2x^2 + 4x - 8 - \frac{6}{x+2}\]Such rational expressions allow us to discern complicated polynomial behaviors and offer insight into asymptotic behavior or simplification opportunities.

Manipulating rational expressions deftly requires an understanding of both polynomial division and equally managing remainders. It lays the groundwork for advanced algebra and calculus topics, leading to complex analysis and practical application in real-world problems.