Problem 13 - Algebra for College Students

Question

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(5 x^{3}-9 x^{2}-3 x-2\right) \div(x-2) $$

Step-by-Step Solution

Answer

Quotient: \(5x^2 + x - 1\), Remainder: \(-4\).

Step 1: Set up Synthetic Division

To perform synthetic division, first, write down the coefficients of the dividend polynomial \(5x^3 - 9x^2 - 3x - 2\), which are \(5, -9, -3, -2\). Place these coefficients in a row. Since the divisor is \(x - 2\), the value to use is \(2\) (the root of the divisor).

Step 2: Begin Synthetic Division Process

Bring down the first coefficient, \(5\), as it is. This starts the division process. Now, multiply \(5\) by \(2\) (the root of the divisor) to get \(10\), and write this value under the next coefficient, \(-9\).

Step 3: Perform Addition in Synthetic Division

Add \(-9\) and \(10\) together to get \(1\). Write \(1\) below the line. Next, multiply \(1\) by \(2\) to get \(2\), and place this value under the next coefficient, \(-3\).

Step 4: Continue Division Process

Add \(-3\) and \(2\) to get \(-1\). Write \(-1\) below the line. Now, multiply \(-1\) by \(2\) to get \(-2\), and write this under the next coefficient, \(-2\).

Step 5: Finalize Remainder Calculation

Add \(-2\) and \(-2\) to get \(-4\). This final value is the remainder of the division operation.

Step 6: Interpret the Result

The result from the synthetic division operation is \(5x^2 + 1x - 1\) with a remainder of \(-4\).

Key Concepts

Polynomial Division

Polynomial division is a method to divide two polynomials. It's akin to numerical division but with variables involved. The process helps in breaking down complex polynomial expressions into simpler components, usually a quotient and a remainder.

• The polynomial we divide is called the "dividend" and the polynomial we divide by is called the "divisor."
• The objective is to determine the quotient, the polynomial that results from the division, and the remainder, which is what's left over.

Polynomial division can be performed manually using methods like long division or synthetic division. Each has its advantages, but synthetic division is often quicker and is used when the divisor is in the form \(x - c\). It's essential to ensure all terms are in place before starting. Missing terms should be represented by a coefficient of zero for simplicity.

Quotient and Remainder

When dividing polynomials, two main results are produced: the quotient and the remainder.

• The quotient is the result of dividing the dividend by the divisor, essentially simplifying the original expression.
• The remainder is the piece of the dividend that isn't completely divisible by the divisor. This is similar to getting a remainder in basic arithmetic division.

In our exercise, after carrying out the synthetic division, the quotient is determined to be \(5x^2 + 1x - 1\) and the remainder is \(-4\). These results highlight the parts of the polynomial that are fully divisible and the part that is not, respectively.
Understanding the relationship between the quotient, remainder, dividend, and divisor is crucial for verifying the accuracy of the division. This is often represented as: \(\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}\).

Coefficient Matching

Coefficient matching is a strategy used in polynomial division that involves aligning and balancing the coefficients of like terms.

• During synthetic division, coefficients of the dividend polynomial are laid out in a sequence.
• As the process proceeds, new coefficients for the quotient are calculated step by step, ensuring they "match" by fulfilling the necessary arithmetic operations with accurate results.

In practical terms, as you run through synthetic division, you repeatedly multiply the divisor's root with the current quotient term. Then, you add this product to the next coefficient in the dividend. This process of systematic errors and adjustments is akin to balancing an equation, ensuring polynomial integrity.

Algebraic Techniques

Algebraic techniques such as synthetic division simplify polynomial division, making it more efficient.

• Synthetic division is best suited for polynomials divided by binomials of the form \(x - c\). It reduces complex algebra into an array-focused, simple arithmetic process.
• This technique bypasses the extensive steps of polynomial long division by focusing only on coefficients, thus, saving time and minimizing errors.

The core idea is that you only handle numbers and simple multiplication and addition. Simplifying polynomials using these algebraic methods helps understand and solve more complex problems by breaking them into basic manageable parts. This way, algebra becomes less intimidating and more approachable, even for those new to the topic.