Problem 22 - 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.) $$ \left(3 x^{3}-5 x^{2}+2 x+3\right) \div(x+2) $$

Step-by-Step Solution

Answer

The quotient is \(3x^2 - 11x + 24\) with remainder \(-45\).

Step 1: Set Up Synthetic Division

Write the coefficients of the dividend polynomial, which is \(3x^3 - 5x^2 + 2x + 3\), as a row: \([3, -5, 2, 3]\). Identify the root of the divisor \(x + 2 = 0\), which is \(x = -2\). This will be used to perform the synthetic division.

Step 2: Begin Synthetic Division

Bring down the first coefficient \(3\) as it is and write it below the line. This begins the process.

Step 3: First Iteration

Multiply the root \(-2\) by the number below the line \(3\), which gives \(-6\). Add this to the next coefficient \(-5\), resulting in \(-11\). Write \(-11\) beneath the line.

Step 4: Second Iteration

Multiply \(-2\) by \(-11\), getting \(22\). Add \(22\) to the next coefficient \(2\), resulting in \(24\). Write \(24\) beneath the line.

Step 5: Third Iteration

Multiply \(-2\) by \(24\), getting \(-48\). Add \(-48\) to the last coefficient \(3\), resulting in \(-45\). Write \(-45\) beneath the line. This is the remainder.

Step 6: Form the Quotient

The numbers under the line, excluding the last remainder \(-45\), represent the coefficients of the quotient: \(3x^2 - 11x + 24\). The complete quotient is \(3x^2 - 11x + 24\) with a remainder of \(-45\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide a polynomial by another polynomial, much like how you divide numbers. It allows us to break down complex algebraic expressions into simpler parts. There are different techniques to perform polynomial division, including long division and synthetic division.

• **Long Division:** This is analogous to numerical long division and involves dividing the terms one by one to obtain the quotient and remainder.
• **Synthetic Division:** A streamlined method suitable when the divisor is a linear polynomial of the form \(x - c\), which is typically more efficient than long division.

Synthetic division is particularly useful for dividing polynomials when the divisor is a straightforward expression like \(x + 2\). It minimizes the complexity by focusing on the coefficients instead of the entire expressions, making the calculations quicker and easier.

Remainder Theorem

The Remainder Theorem is a valuable principle in algebraic expressions. It provides an efficient way to find the remainder when a polynomial is divided by a linear divisor. The theorem states that when a polynomial \(f(x)\) is divided by \((x - c)\), the remainder is \(f(c)\).
This means if you substitute \(c\) into the polynomial, the result is the remainder.

• For example, dividing \(3x^3 - 5x^2 + 2x + 3\) by \(x + 2\) using synthetic division results in a remainder of \(-45\).
• By substituting \(-2\) (since \(x + 2 = 0\) gives \(x = -2\)) into the polynomial, you can verify that the remainder is indeed \(-45\).

Using the Remainder Theorem through synthetic division is not only a time-saver but also provides a quick verification method for the remainder.

Divisor Polynomial

A divisor polynomial is a polynomial by which another polynomial is divided. In the context of synthetic division, the divisor is typically a simple linear polynomial like \(x + 2\). This simplicity is key to performing synthetic division efficiently.
To use synthetic division, the divisor polynomial must be transformed into the form \(x - c\):

• For \(x + 2\), rewrite it as \(x - (-2)\), identifying \(-2\) as the value used in synthetic division.
• This transformation allows us to focus on the coefficients as we carry out synthetic division.

By reducing the polynomial to its core components, synthetic division utilizes the structure of the divisor polynomial for rapid calculations.

Algebraic Expressions

Algebraic expressions are built using constants, variables, and coefficients and are combined through operations like addition, subtraction, multiplication, and division. They form the foundation for polynomial equations.

• In expressions such as \(3x^3 - 5x^2 + 2x + 3\), each term represents a part of the overall expression.
• The mechanics of these expressions are crucial during division, especially when arranging terms appropriately for synthetic division.
• Simplification of these expressions through techniques like synthetic division helps in problem-solving and further algebraic manipulation.

Understanding how to manipulate algebraic expressions is vital in mastering more complex mathematical concepts, and synthetic division provides a practical approach to simplifying these expressions efficiently.