Problem 14 - Intermediate Algebra

Question

Use synthetic division to perform each division. See Example 1. $$ \left(4 x^{2}-5 x-6\right) \div(x-2) $$

Step-by-Step Solution

Answer

The result of the division is \(4x + 3\) with no remainder.

Step 1: Set Up Synthetic Division

Write down the coefficients of the dividend polynomial \(4x^2 - 5x - 6\) which are \(4, -5, -6\). The divisor \(x - 2\) has a root \(2\), so use that as the synthetic divisor.

Step 2: Perform the Division

1. Bring down the leading coefficient \(4\) from the dividend to start the quotient.2. Multiply it by the divisor root \(2\) and place the result \(8\) under the next coefficient \(-5\).3. Add \(-5 + 8 = 3\). Write \(3\) below the line.4. Repeat the process: multiply the new result \(3\) by \(2\) to get \(6\) and place it under \(-6\).5. Add \(-6 + 6 = 0\). Write \(0\) below the line, indicating no remainder.

Step 3: Write Down the Quotient

The synthetic division process results in row \(4, 3\) with no remainder. This corresponds to the polynomial \(4x + 3\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. When we have a dividend and a divisor, it works similarly to arithmetic division. The goal is to simplify the expression. This is often needed in algebra when working with higher degree polynomials.

In the example provided, we are dividing \(4x^2 - 5x - 6\) by \(x - 2\). The polynomial division can be done either using long division or synthetic division. Synthetic division is often preferred for cases where the divisor is of the form \(x - c\), as it provides a more streamlined and quicker process.

Coefficients

Coefficients are the numbers in front of the variables in a polynomial. They play a crucial role in polynomial operations, including division. When using synthetic division, the coefficients are the primary elements used in the calculations.

In the exercise, we break down the polynomial \(4x^2 - 5x - 6\) into its coefficients: \(4, -5, -6\). These numbers are essential for setting up the synthetic division process as they represent the "weight" or "strength" of each corresponding term in the polynomial. Without correctly identifying these coefficients, the synthetic division would be inaccurate, leading to incorrect results.

Remainder Theorem

The Remainder Theorem is a handy tool in polynomial division. It states that if a polynomial \(f(x)\) is divided by \(x - c\), the remainder of this division is \(f(c)\).

Applying the theorem in synthetic division, it helps us quickly verify results. In our example, the synthetic division of \(4x^2 - 5x - 6\) by \(x - 2\) resulted in a remainder of zero. This indicates that \(x = 2\) is a root of the polynomial, meaning it perfectly divides the polynomial with no remainder.

• In practical terms, having a remainder of zero confirms that the factor \(x - 2\) divides the polynomial completely.

Quotient

The quotient in polynomial division is the result obtained when the dividend is divided by the divisor. In simpler terms, it is the "answer" to the division.

During the synthetic division process, we identify the quotient by tracking the numbers left below the line after the operations are completed. In the provided solution, after synthetic division, we obtain \(4, 3\). This relates to the polynomial quotient \(4x + 3\).

• The quotient describes the remaining polynomial after the division process, helping us understand how many times the divisor fits into the dividend, excluding the remainder.

Understanding the quotient is essential as it tells us the simplified version of the original division expression.