Problem 9 - College Algebra with Modeling and Visualization

Question

Divide the first polynomial by the second. State the quotient and remainder. $$x^{3}-2 x^{2}-5 x+6 \quad\quad\quad x-3$$

Step-by-Step Solution

Answer

Quotient: \(x^2 + x - 2\); Remainder: \(0\).

Step 1: Setup Polynomial Division

Align the terms of the polynomial \(x^3 - 2x^2 - 5x + 6\) in descending order and prepare to divide them by \(x - 3\). We will use synthetic division here.

Step 2: Perform Synthetic Division

Use synthetic division, placing \(3\) in the synthetic division bracket (as we are dividing by \(x - 3\)). Write the coefficients of the dividend: \(1, -2, -5, 6\). Start the synthetic division process.- Bring down the leading coefficient, \(1\).- Multiply \(1\) by \(3\) and add to the next coefficient: \(-2 + 3 \times 1 = 1\).- Multiply the result \(1\) by \(3\) and add to \(-5\): \(-5 + 3 \times 1 = -2\).- Multiply the result \(-2\) by \(3\) and add to \(6\): \(6 + 3 \times -2 = 0\).The final row gives us the coefficients of the quotient \(1, 1, -2\) and a remainder \(0\).

Step 3: Write the Quotient and Remainder

Translate the synthetic division result into the polynomial quotient and remainder. The quotient is \(x^2 + x - 2\) and the remainder is \(0\).

Step 4: Verify the Result

Multiply the quotient \(x^2 + x - 2\) by \(x - 3\) and verify that the result equals the original polynomial \(x^3 - 2x^2 - 5x + 6\).\[(x^2 + x - 2)(x - 3) = x^3 - 3x^2 + x^2 - 3x - 2x + 6 = x^3 - 2x^2 - 5x + 6\]Since the product equals the original polynomial with no remainder, our quotient and remainder are verified.

Key Concepts

Synthetic Division

Synthetic Division is a streamlined form of polynomial division, specifically designed for dividing polynomials by linear expressions of the form \(x - c\). It simplifies the division process by focusing only on the coefficients of the polynomial, allowing for quicker calculations without cumbersome variables. To perform Synthetic Division, follow these steps:

• Write down the coefficients of the dividend polynomial. For our exercise, these are \(1, -2, -5, 6\).
• Place the root of the divisor, \(c\), into a bracket. In our case, since we are dividing by \(x - 3\), \(c\) is \(3\).
• Bring down the leading coefficient of the dividend directly below the bracket.
• Multiply this number by \(c\) and add it to the next coefficient.
• Continue this process down the line of coefficients.

This method provides both the coefficients of the quotient and the remainder in a straightforward manner. It's especially handy for its simplicity and efficiency.

Remainder Theorem

The Remainder Theorem is a powerful tool that connects polynomial division with polynomial functions. According to the theorem, when a polynomial \(f(x)\) is divided by a linear divisor \(x - c\), the remainder of this division is equal to \(f(c)\). Here's how it relates to our problem:

• We divided \(x^3 - 2x^2 - 5x + 6\) by \(x - 3\).
• Synthetic Division yielded a remainder of \(0\).
• According to the theorem, this means \(f(3) = 0\). This confirms that \(x - 3\) is indeed a factor of the dividend polynomial.

The Remainder Theorem is not only a time-saver but also helps in revealing useful properties and factor relationships in polynomials.

Quotient and Remainder Verification

Quotient and Remainder Verification is a critical step to ensure the division process was completed correctly. Verification involves checking if multiplying the quotient by the divisor and adding the remainder returns the original polynomial. In our example, the calculated quotient was \(x^2 + x - 2\) and the remainder was \(0\). Let's verify: First, multiply the quotient and the divisor: \[(x^2 + x - 2)(x - 3) = x^3 - 3x^2 + x^2 - 3x - 2x + 6\] Simplify the expression: \[x^3 - 2x^2 - 5x + 6\] This confirms the product matches the original polynomial. Since our remainder is \(0\), there is nothing more to add. This verification process ensures accuracy and correctness in polynomial division, building confidence in the solution provided.