Problem 9 - Algebra and Trigonometry

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 $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ \(P(x)=4 x^{2}-3 x-7, \quad D(x)=2 x-1\)

Step-by-Step Solution

Answer

Quotient: \(2x - 1\); Remainder: \(-\frac{9}{2}\). Divide: \(2x - 1 + \frac{-\frac{9}{2}}{2x - 1}\).

Step 1: Set Up Synthetic Division

First, determine the zero of the divisor \(D(x) = 2x - 1\). Set \(2x - 1 = 0\) so \(x = \frac{1}{2}\). Now, for the polynomial \(P(x) = 4x^2 - 3x - 7\), write down the coefficients: 4, -3, and -7. Set up the synthetic division table using \(\frac{1}{2}\) as the divisor.

Step 2: Synthetic Division Process

Bring down the leading coefficient (4) to the bottom row. Multiply this by the divisor \(\frac{1}{2}\) and place the result under the next coefficient (-3). Add this to -3 to get the new entry. Repeat the process: multiply the new entry by \(\frac{1}{2}\), place under the following coefficient, and add. Continue until all coefficients are processed.

Step 3: Identify Quotient and Remainder

The bottom row of synthetic division gives the coefficients of the quotient and remainder. In this case, you end up with coefficients for a linear quotient \(Q(x) = 2x - 1\) and a remainder \(R(x) = -\frac{9}{2}\).

Step 4: Express in Required Form

Express the division result in the required form as: \[\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} = 2x - 1 + \frac{-\frac{9}{2}}{2x - 1}\].

Key Concepts

Synthetic Division

Synthetic division is a shorthand method of dividing polynomials when the divisor is of the form \(x - c\). It simplifies the process by using only the coefficients of the polynomials.
To perform synthetic division, you start by identifying the zero of the divisor, which we calculated as \(\frac{1}{2}\) from \(2x - 1 = 0\). This number is used to repeatedly multiply and add through the coefficients of the dividend.

• Only the coefficients of the polynomial are used in this method, which reduces potential errors and time spent on computation.
• The method is particularly useful for divisions with simpler linear divisors.

Through synthetic division, we obtained the quotient \(2x - 1\) and the remainder \( -\frac{9}{2} \). This method is efficient and powerful in simplifying polynomial division.

Long Division

Long division for polynomials is like the long division process for numbers, involving repeated division, multiplication, and subtraction.
This method requires dividing the highest degree term of the dividend by the highest degree term of the divisor at each step.

• Each division step creates a new term in the quotient, which is multiplied by the divisor and subtracted from the current dividend.
• This process is repeated until the remainder becomes of lower degree than the divisor.

Even though it is straightforward, long division can be time-consuming and prone to errors as it involves managing multiple terms through each cycle of the process.

Quotient and Remainder

The result from dividing one polynomial by another consists of two main parts: the quotient \(Q(x)\) and the remainder \(R(x)\).
The quotient is the result of the division; a polynomial whose degree is less than or equal to the degree of the dividend minus the degree of the divisor.

• The remainder is what's left after the division has processed all terms of the dividend and is of lesser degree than the divisor.
• The division statement can be written in the form \(\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}\), where \(P(x)\) is the original polynomial and \(D(x)\) is the divisor.

In our example, the quotient is \(2x - 1\), and the remainder is \(-\frac{9}{2}\), fitting perfectly in the division statement structure.

Polynomial Coefficients

Coefficients are the numerical factors in the polynomial terms. They play a crucial role in both synthetic and long division.
In synthetic division, the process deals directly with these coefficients, making it simpler and cleaner. The polynomial \(P(x) = 4x^2 - 3x - 7\) has coefficients 4, -3, and -7.

• Coefficients dictate the values that will be multiplied and added during the synthetic division process.
• They determine the outcomes at each stage of the division and are essential in forming the coefficient of the quotient polynomial.

By understanding and managing coefficients properly, you can more effectively carry out polynomial division.