Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(9 x^{3}-9 x^{2}+18 x+5\right) \div(3 x-1) $$

Step-by-Step Solution

Answer

The quotient is \(9x^2 - 6x + 16\), and the remainder is \(\frac{31}{3}\).

Step 1: Reformat the Problem

The given polynomial division is \((9x^3 - 9x^2 + 18x + 5) \div (3x - 1)\). We will first reformat the divisor by dividing it by the coefficient of the linear term. So, \((3x - 1)\) becomes \(x - \frac{1}{3}\). Now the division is \((9x^3 - 9x^2 + 18x + 5) \div \left(x - \frac{1}{3}\right)\).

Step 2: Set Up for Synthetic Division

Write down the coefficients of the dividend polynomial, which are 9, -9, 18, and 5. The root of the divisor \(x - \frac{1}{3}\) is \(\frac{1}{3}\). This root is what we'll use for synthetic division.

Step 3: Perform the Synthetic Division

Draw a synthetic division setup with the coefficients \(9, -9, 18, 5\) and the root \(\frac{1}{3}\). Start by bringing down the first coefficient (9) unchanged:- Multiply 9 by \(\frac{1}{3}\) and write the result under the next coefficient.- Add this result to the next coefficient (-9).- Repeat this process for each coefficient: multiply, then add to the next original coefficient.

Step 4: Calculate Step-by-Step

1. Bring down 9.2. Multiply 9 by \(\frac{1}{3}\) to get 3, add to -9 to get -6.3. Multiply -6 by \(\frac{1}{3}\) to get -2, add to 18 to get 16.4. Multiply 16 by \(\frac{1}{3}\) to get \(\frac{16}{3}\), add to 5. - This gives us the remainder: \(5 + \frac{16}{3} = \frac{15}{3} + \frac{16}{3} = \frac{31}{3}\). The quotient coefficients are 9, -6, and 16, with a remainder \(\frac{31}{3}\).

Step 5: Form the Quotient and Remainder

The quotient is obtained by using the coefficients from synthetic division as powers of descending order. Thus, the quotient polynomial is \(9x^2 - 6x + 16\). The remainder is \(\frac{31}{3}\), so the solution to the division is: \[9x^2 - 6x + 16 + \frac{\frac{31}{3}}{x - \frac{1}{3}} \]

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. This process is similar to long division with numbers but involves variables. When dealing with polynomial division, the goal is to divide the dividend (the polynomial being divided) by the divisor (the polynomial you are dividing by) to find the quotient and remainder. In this specific exercise, we are dividing the polynomial \(9x^3 - 9x^2 + 18x + 5\) by \(3x - 1\).
To fully grasp polynomial division, it's helpful to understand that:

• The dividend is the polynomial you are dividing.
• The divisor is the polynomial you are dividing by.
• The quotient is the result of the division.
• The remainder is what's left after division.

Keep in mind that dividing polynomials is about simplifying and approximating solutions, much like with numbers.

Quotient and Remainder

The quotient and remainder are crucial components of polynomial division. The quotient is essentially the result of the division when you divide the dividend by the divisor. Meanwhile, the remainder is what's left over after the division process is completed. In polynomial terms, the remainder indicates that the divisor does not fully factor into the dividend.
In our exercise, after using synthetic division on \((9x^3 - 9x^2 + 18x + 5) \div (3x - 1)\), we found that the quotient is \(9x^2 - 6x + 16\). The remainder is \(\frac{31}{3}\). So, the polynomial division can be expressed as:
\[9x^2 - 6x + 16 + \frac{\frac{31}{3}}{x - \frac{1}{3}}\] Understanding the quotient and remainder helps validate the division process. If there is a remainder, it means the divisor is not a perfect factor of the dividend.

Synthetic Division Setup

Synthetic division is an efficient method used to divide polynomials, particularly useful when the divisor is a linear polynomial of the form \(x - c\). The setup involves arranging the coefficients of the dividend and applying the root of the divisor. Here, with a divisor of \(3x - 1\), we converted it to \(x - \frac{1}{3}\) for simplicity.
To set up synthetic division:

• List the coefficients of the polynomial dividend: 9, -9, 18, and 5.
• Determine the root from the divisor, \(x - \frac{1}{3}\), which is \(\frac{1}{3}\).

The synthetic division is then performed by iteratively multiplying and adding these coefficients. This approach streamlines calculations and reduces errors, making polynomial division quicker and easier.

Polynomial Coefficients

Polynomial coefficients play a critical role in synthetic division. These numbers provide the values needed to perform calculations that yield the quotient and remainder. Each term in a polynomial has a coefficient, which is the number in front of the variable.
In our problem, the polynomial \(9x^3 - 9x^2 + 18x + 5\) has coefficients:

• 9 for \(x^3\)
• -9 for \(x^2\)
• 18 for \(x\)
• 5 as the constant term

During synthetic division, these coefficients are manipulated with the divisor's root to methodically simplify the polynomial. Understanding coefficients is crucial as they affect the outcome of each multiplication and addition step, determining the accuracy of the quotient and remainder.