Problem 12 - Precalculus Mathematics for Calculus

Question

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ $$P(x)=6 x^{3}+x^{2}-12 x+5, \quad D(x)=3 x-4$$

Step-by-Step Solution

Answer

The quotient is \(2x^2 + 3x\) and the remainder is \(\frac{5}{3x-4}\).

Step 1: Set Up the Division

To divide the polynomial \(P(x) = 6x^3 + x^2 - 12x + 5\) by \(D(x) = 3x - 4\), we'll use synthetic division because \(D(x)\) is a first degree polynomial. First, solve \(3x - 4 = 0\) to find the value used in synthetic division. The zero is \(x = \frac{4}{3}\). Write the coefficients of \(P(x)\), which are \([6, 1, -12, 5]\), and set up the synthetic division by placing \(\frac{4}{3}\) on the left.

Step 2: Perform Synthetic Division

In synthetic division, bring down the leading coefficient, \(6\). Multiply it by \(\frac{4}{3}\) and write the result under the next coefficient. Add these to get the new coefficient. Repeat the same process for the entire polynomial:- Bring down \(6\).- Multiply: \(\frac{4}{3} \times 6 = 8\). Add to the next coefficient: \(1 + 8 = 9\).- Multiply: \(\frac{4}{3} \times 9 = 12\). Add to the next: \(-12 + 12 = 0\).- Multiply: \(\frac{4}{3} \times 0 = 0\). Add to the next: \(5 + 0 = 5\).The final row is \([6, 9, 0, 5]\), where \(6x^2 + 9x + 0\) is the quotient and \(5\) is the remainder.

Step 3: Write the Quotient and Remainder

The result of the synthetic division gives us the quotient \(Q(x) = 2x^2 + 3x\) and the remainder \(R(x) = 5\). Therefore, the expression for \(\frac{P(x)}{D(x)}\) is:\[ \frac{P(x)}{D(x)} = 2x^2 + 3x + \frac{5}{3x - 4} \]

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing polynomials, primarily used when the divisor is a first-degree polynomial of the form \( x - c \). It streamlines the process of polynomial division, making it faster and simpler compared to long division, especially when you're calculating manually. To begin with synthetic division, here's what you do:

• First, identify the value of \( c \) by setting the divisor equal to zero and solving for \( x \). For example, if your divisor is \( 3x - 4 \), then \( x = \frac{4}{3} \).
• Next, list the coefficients of the dividend polynomial. For \( P(x) = 6x^3 + x^2 - 12x + 5 \), the coefficients are \([6, 1, -12, 5]\).
• Position \( c \) to the left of the setup and the coefficients in a row. Begin the division process by bringing down the first coefficient directly below the line.
• Multiply the result by \( c \) and add it to the next coefficient. Repeat this step for each coefficient.

It’s a step-by-step mechanism where each multiplication and addition builds on the previous result. The row of numbers at the bottom gives you the coefficients of the quotient, and the last number is the remainder.

Long Division of Polynomials

Long division of polynomials resembles the familiar arithmetic long division process. Although more lengthy than synthetic division, it is a very effective method, especially when dealing with higher-degree divisors.

Here's how you follow the steps for long division of polynomials:

• Divide the leading term of the dividend by the leading term of the divisor. Write this value above the division line.
• Multiply the entire divisor by this term and subtract the result from the original polynomial.
• Bring down the next term of the dividend if necessary, and repeat the process.
• Continue until the degree of the remainder is less than the degree of the divisor.

The terms you wrote above the division line form the quotient, and whatever is left at the bottom is the remainder. This method allows for a structured and logical way to break down more complex polynomial divisions, ensuring that you handle each part correctly.

Quotient and Remainder Theorem

The Quotient and Remainder Theorem is a fundamental concept in polynomial division. It asserts that for any two polynomials \( P(x) \) and \( D(x) \), where \( D(x) \) is not zero, you can express the ratio of these polynomials as:\[ P(x) = D(x) \cdot Q(x) + R(x) \]

• Here, \( Q(x) \) is the quotient obtained when \( P(x) \) is divided by \( D(x) \).
• The remainder \( R(x) \) has a degree that is less than the degree of \( D(x) \).
• This theorem simplifies understanding polynomial division by clearly defining the roles of the quotient and remainder.

This concept is not just a theoretical framework but also a practical tool that provides clear expectations when dividing any polynomial. The expression \( \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} \) allows us to see how the original polynomial was composed of these distinct divisible parts.