Problem 6 - Algebra and Trigonometry

Question

\(3-8=\) Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\) $$P(x)=4 x^{3}+7 x+9, \quad D(x)=2 x+1$$

Step-by-Step Solution

Answer

The quotient is \(4x^2 - 2x + 8\) and the remainder is 5.

Step 1: Set Up the Division

Write down the coefficients for the polynomial \(P(x) = 4x^3 + 0x^2 + 7x + 9\) since there is no \(x^2\) term, and the divisor \(D(x) = 2x + 1\). We want to use synthetic division, so we should rearrange \(D(x)\) as \(2x + 1\). The zero of \(D(x)\) is \(-\frac{1}{2}\).

Step 2: Synthetic Division Setup

For synthetic division, set up the coefficients of \(P(x)\) as a row: \(4, 0, 7, 9\). Write \(-\frac{1}{2}\) on the left as the divisor. Leave space underneath these values for the calculations.

Step 3: Start the Division Process

Bring down the leading coefficient (4) to the row below. Multiply \(4\) by \(-\frac{1}{2}\) to get \(-2\). Add this to the next coefficient (0) to get \(-2\).

Step 4: Continue the Division

Multiply \(-2\) by \(-\frac{1}{2}\) to get \(1\). Add this to the next coefficient (7) to get \(8\).

Step 5: Finalise the Division

Multiply \(8\) by \(-\frac{1}{2}\) to get \(-4\). Add this to the next coefficient (9) to get \(5\). This last number represents the remainder.

Step 6: Write the Result as a Polynomial

The result of the division gives the quotient coefficients \([4, -2, 8]\), representing the polynomial \(Q(x) = 4x^2 - 2x + 8\), and the remainder is \(R(x) = 5\). Therefore, \(P(x) = D(x) \cdot Q(x) + R(x) = (2x + 1)(4x^2 - 2x + 8) + 5\).

Key Concepts

Understanding Synthetic Division

Synthetic division is a simplified method of dividing polynomials that is especially useful when dividing by a linear divisor of the form \(x - c\). It streamlines the process and reduces the amount of calculations required by working only with the coefficients.
Here's how it works:

• First, identify the coefficients of the polynomial you're dividing. Make sure to include zeros for any missing terms. For example, if the polynomial is \(4x^3 + 7x + 9\), write this as \(4, 0, 7, 9\).
• The divisor \(D(x)\) is in the form \(2x + 1\), from which we find the zero by solving \(2x + 1 = 0\). The solution is \(x = -\frac{1}{2}\).
• Begin the synthetic division by bringing down the first coefficient to the line below. This starts the process of multiplying and adding through the row of coefficients.

This method is faster than long division because it bypasses polynomial expressions, focusing instead on numerical operations with coefficients.

What Are Polynomials?

Polynomials are expressions consisting of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents of variables. They are foundational elements in algebra and calculus, found often in mathematical modeling and problem-solving situations. Some key features include:

• Terms: Individual elements in the polynomial, such as \(4x^3\) or \(7x\).
• Coefficients: Numbers multiplying the variable parts, such as 4 in \(4x^3\).
• Degree: The highest power of the variable in the expression, which helps to determine the behavior of the polynomial.

In the example \(4x^3 + 7x + 9\), it's a cubic polynomial because the highest power of \(x\) is 3. Understanding how to manipulate polynomials, including division, is crucial for solving many algebraic problems.

Exploring the Remainder Theorem

The Remainder Theorem gives us a valuable tool for understanding division of polynomials. It states that when you divide a polynomial \(P(x)\) by a linear divisor \(x - c\), the remainder of this division is equal to \(P(c)\). This theorem simplifies verification of results obtained through synthetic or long division.
Here's how you can apply it:

• After using synthetic division, the last number in your row of results is the remainder of the division.
• Substitute \(c\) into your original polynomial to confirm that this remainder is accurate. For example, substituting \(-\frac{1}{2}\) into the polynomial \(4x^3 + 7x + 9\) should yield 5, confirming our remainder.
• The theorem not only validates your result but also allows evaluations of polynomial values more efficiently.

Using the Remainder Theorem, you gain confidence in your division results, knowing that the value calculated for the remainder is indeed correct without conducting the full division process again.