Problem 11 - Algebra and Trigonometry

Question

\(9-14\) . 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

The quotient is \( 2x - 1 \) with remainder \( -7.5 \). Divide to form \( \frac{4x^2-3x-7}{2x-1} = (2x-1) + \frac{-7.5}{2x-1} \).

Step 1: Set Up the Division

To divide the polynomial \( P(x) = 4x^2 - 3x - 7 \) by \( D(x) = 2x - 1 \), we will use synthetic division. First, we solve \( 2x-1 = 0 \) to get \( x = \frac{1}{2} \). We will use this root for synthetic division. Arrange the coefficients of \( P(x) \) in order: 4, -3, and -7.

Step 2: Perform Synthetic Division

Write down the coefficients 4, -3, -7. Bring down the leading coefficient 4 as it is. Multiply 4 by \( \frac{1}{2} \) (the root), getting 2, and add this to the next coefficient -3 to get -1. Multiply \( -1 \) by \( \frac{1}{2} \), resulting in \( -\frac{1}{2} \), and add to -7 to obtain \( -7.5 \). The quotient is thus \( 4x - 1 \), and the remainder is \( -7.5 \).

Step 3: Express Quotient and Remainder in Desired Form

Express the result in the form \( Q(x) + \frac{R(x)}{D(x)} \). Here, \( Q(x) = 2x - 1 \) and \( R(x) = -7.5 \). Thus, in the form requested, we have \[ \frac{4x^2 - 3x - 7}{2x - 1} = (2x - 1) + \frac{-7.5}{2x - 1} \].

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial division. It is particularly useful when dividing by a linear polynomial of the form \( x - c \). Instead of using long division's more tedious steps, synthetic division offers a quick way to find the quotient and remainder.Here's how it works:

• First, solve \( 2x - 1 = 0 \) to find the value of \( x \) that will be used, which in this case is \( x = \frac{1}{2} \).
• Write down the coefficients of the polynomial \( P(x) \), which are 4, -3, and -7.
• Bring down the first coefficient, and using the root \( \frac{1}{2} \), multiply and add sequentially through the coefficients.
• The last number yielded in the process is the remainder, while the other numbers form the coefficients of the quotient.

With practice, synthetic division becomes an invaluable tool for quickly dividing polynomials.

Long Division

Long division of polynomials is similar to long division of numbers. It's methodical and works for any divisor regardless of its degree. The process involves:

• Dividing the highest degree term of the dividend by the highest degree term of the divisor.
• Multiply the entire divisor by the result and subtract it from the original dividend.
• Repeat the steps above with the new polynomial formed until you reach a remainder smaller in degree than the divisor.

Though more complex than synthetic division, long division is a reliable method, especially when dealing with divisors that are not linear.

Remainder Theorem

The Remainder Theorem states that the remainder of dividing a polynomial \( P(x) \) by a linear factor \( x - c \) is \( P(c) \). That means if you substitute \( c \) into the polynomial, the result will be the remainder of the division.In our case:

• We use \( x = \frac{1}{2} \) from dividing by \( 2x - 1 \).
• Solve \( P\left(\frac{1}{2}\right) \) to find the remainder, confirming the results of synthetic or long division.

This theorem provides a quick check for the remainder if calculations need verifying.

Polynomial Quotient

The quotient of a polynomial division is what you get after eliminating the divisor's highest terms from the dividend. It represents simpler polynomial terms that approximate the behavior of the original polynomial divided by the divisor.Here's how to conceptualize:

• Using our division, the quotient \( Q(x) \) was determined to be \( 2x - 1 \).
• This quotient, when multiplied back by the divisor \( 2x - 1 \), minus the remainder, will reproduce the original polynomial.

The polynomial quotient provides insights into the polynomial's structure and can be used in further algebraic manipulations.