Problem 17 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(4 x^{3}-12 x^{2}-5 x-1\right) \div(2 x+1) $$

Step-by-Step Solution

Answer

The quotient is \(4x^2 - 14x + 2\) with a remainder of \(-2\).

Step 1: Identify the Divisor

First, identify the divisor in the given expression. Here, the divisor is \(2x + 1\). We need to set it equal to zero to find the number to use in synthetic division. Set \(2x + 1 = 0\), which gives \(x = -\frac{1}{2}\). This value will be used in synthetic division.

Step 2: Arrange the Coefficients

List the coefficients of the dividend polynomial \(4x^3 - 12x^2 - 5x - 1\). These coefficients are \(4\), \(-12\), \(-5\), and \(-1\). We will use these in synthetic division.

Step 3: Set Up Synthetic Division

Draw the synthetic division setup. Place \(-\frac{1}{2}\) (from Step 1) to the left of the division bar. Write the coefficients \(4\), \(-12\), \(-5\), and \(-1\) in a row next to the division bar.

Step 4: Begin the Division Process

Bring down the leading coefficient \(4\) directly under the line. Multiply \(4\) by \(-\frac{1}{2}\) to get \(-2\). Write \(-2\) under the next coefficient \(-12\). Add \(-12\) and \(-2\) to get \(-14\). Continue this process.

Step 5: Complete the Division

Continue multiplying and adding. Multiply \(-14\) by \(-\frac{1}{2}\) to get \(7\). Add this to \(-5\) to get \(2\). Multiply \(2\) by \(-\frac{1}{2}\) to get \(-1\). Add this to \(-1\) to get \(-2\). We will obtain remaining coefficients \(4, -14, 2\) and remainder \(-2\).

Step 6: Write the Quotient and Remainder

The result of the synthetic division gives us a quotient polynomial and a remainder. The coefficients \(4, -14, 2\) represent the polynomial \(4x^2 - 14x + 2\). The remainder is \(-2\). Thus, the quotient is \(4x^2 - 14x + 2\) with a remainder of \(-2\).

Key Concepts

Polynomial Division

When we talk about polynomial division, it's much like long division that you've learned in arithmetic. We're dividing one polynomial by another, and we can use methods like synthetic division to make it simpler and quicker.
Let's break this down:

• **Dividend**: The polynomial you are dividing. In this exercise, it's \(4x^3 - 12x^2 - 5x - 1\).
• **Divisor**: The polynomial by which you are dividing. Here, it's \(2x + 1\).
• **Quotient**: The result of the division.
• **Remainder**: What's left over after the division.

Synthetic division simplifies this process and only involves the coefficients of the polynomials instead of the whole expressions. It works well when dividing by a linear polynomial like \(2x + 1\).
In the exercise, synthetic division allows us to quickly find the quotient: \(4x^2 - 14x + 2\), and remainder: \(-2\). This method saves you from more complex traditional division.

Remainder Theorem

The Remainder Theorem is an essential part of understanding polynomial division. It tells you about the connection between the divisor and the remainder.
Simply put, when you divide a polynomial by \(x - a\), the remainder of this division is the same as the polynomial evaluated at \(a\). In other words:\[P(a) = ext{Remainder}\]Applying this to our exercise:

• The divisor used is \(2x + 1\), which converts to \(x = -\frac{1}{2}\) for synthetic division.
• The remainder \(-2\) is what you would get if you substituted \(-\frac{1}{2}\) into the original polynomial \(4x^3 - 12x^2 - 5x - 1\).

Knowing this can serve as a double-check for your division or a way to evaluate polynomials quickly at a specific point.

Quotient and Remainder

Understanding the quotient and remainder in polynomial division is crucial because they tell you how one polynomial relates to another through division.
Think of the process like this:When you divide, the quotient tells you how many times the divisor fits into the dividend, while the remainder tells you what's left over. This relationship can be expressed as:\[ ext{Dividend} = ext{Divisor} imes ext{Quotient} + ext{Remainder}\]For the exercise, we applied synthetic division:

• The quotient polynomial we obtained was \(4x^2 - 14x + 2\).
• The remainder was \(-2\).

Substituting these into the equation confirms our division:\[4x^3 - 12x^2 - 5x - 1 = (2x + 1)(4x^2 - 14x + 2) - 2\]This shows how the original polynomial breaks down when divided, giving insight into its structure and factors.