Problem 30 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(x^{3}-8\right) \div(x-4) $$

Step-by-Step Solution

Answer

Quotient is \(x^2 + 4x + 16\); remainder is 56.

Step 1: Identify the Division Method

Given the expression \(x^{3} - 8\) divided by \(x-4\), we recognize that this is a polynomial division problem. Since \(x-4\) is a linear divisor, synthetic division is the most efficient method to use here.

Step 2: Setup for Synthetic Division

For synthetic division, we use the zero of the divisor \(x-4\), which is 4. We will divide the coefficients of \(x^{3} - 8\). Since the polynomial \(x^{3} - 8\) is missing the \(x^{2}\) and \(x\) terms, we write the coefficients as \(1, 0, 0, -8\).

Step 3: Perform the Synthetic Division

1. Bring down the leading coefficient 1. 2. Multiply 4 (the root of \(x-4\)) by 1, add the result to the next coefficient (0) to get 4. 3. Multiply 4 by the result (4) to get 16, and add this to the next coefficient (0) to yield 16. 4. Multiply 4 by 16 to get 64, add this to the last coefficient (-8) to obtain 56, which is the remainder.

Step 4: Write the Quotient and the Remainder

The result from synthetic division shows the quotient is \(x^{2} + 4x + 16\) and the remainder is 56. Thus, the division yields:\[\frac{x^{3}-8}{x-4} = x^{2} + 4x + 16 + \frac{56}{x-4}\]

Key Concepts

Synthetic Division

Synthetic division is a straightforward method for dividing polynomials, especially when the divisor is a linear expression like \(x-4\). Generally, synthetic division is quicker and more efficient than long division. Instead of dealing with variables, we focus on the coefficients. This method is particularly beneficial when dealing with higher-degree polynomials.

• Identify the zero of the divisor. For \(x-4\), the zero is 4.
• Set up the coefficients of the dividend. For \(x^{3}-8\), which is missing \(x^{2}\) and \(x\) terms, use \(1, 0, 0, -8\).
• Perform the synthetic division stepwise, multiplying and adding the coefficients.

The process concludes with a quotient and a remainder, providing a concise and simple result of the division without the extended hassle of traditional polynomial division.

Polynomial Quotient

The quotient in polynomial division reflects the simplified expression resulting from the division of the original polynomial by the divisor. After employing synthetic division on the expression \(x^{3} - 8\) by the divisor \(x - 4\), the quotient is determined to be \(x^{2} + 4x + 16\).

• The quotient represents the polynomial result of the division operation excluding the remainder.
• It shows how many times the divisor fits into the dividend, without overstepping the remainder.

The quotient is critical because it indicates the simplified polynomial formed when part of the polynomial is divided evenly by the divisor.

Linear Divisor

A linear divisor is a polynomial of degree one, such as \(x - 4\). Linear divisors are simple, having the general form \(ax + b\) with \(a\) and \(b\) being constants. They are particularly easy to handle in division problems.

• Linear divisors simplify the division process, especially with synthetic division.
• They have one root which is used during synthetic division.

In the given exercise, using the linear divisor \(x-4\), we first determine its root, which is 4, and this value is crucial for the synthetic division process to proceed smoothly.

Remainder in Division

The remainder in polynomial division is what's left over after the division process when the divisor doesn't fit perfectly into the dividend. It's similar to the remainder you're familiar with in arithmetic division, but for polynomials.
For the division of \(x^{3} - 8\) by \(x-4\), the remainder found through synthetic division is 56.

• The remainder is added as part of the final answer in the form \(\frac{remainder}{divisor}\).
• It's vital because it quantifies what's not evenly divisible by the divisor.

In the final result \(x^{2} + 4x + 16 + \frac{56}{x-4}\), 56 represents the remainder, indicating a little portion of the original polynomial that isn't captured by the full quotient.