Problem 44 - College Algebra with Modeling and Visualization

Question

Use synthetic division to divide the first polymomial by the second. $$=x^{5}+\frac{1}{4} x^{4}-x^{3}-\frac{1}{4} x^{2}+3 x-\frac{5}{4} \quad x+\frac{1}{4}$$

Step-by-Step Solution

Answer

Quotient: \(x^4 - x^2 + 3\), Remainder: \(-2\).

Step 1: Set up Synthetic Division

Write down the coefficients of the dividend polynomial, which are: \(1, \frac{1}{4}, -1, -\frac{1}{4}, 3, -\frac{5}{4}\). The divisor is \(x + \frac{1}{4}\), so we use the opposite of the constant term, \(-\frac{1}{4}\), for synthetic division.

Step 2: Bring Down the Leading Coefficient

Bring down the first coefficient, \(1\), as it is, below the line.

Step 3: Multiply and Add

Multiply the number just written below the line (\(1\)) by \(-\frac{1}{4}\) (the identified root from the divisor) and write the result \(-\frac{1}{4}\) under the next coefficient (\(\frac{1}{4}\)) of the dividend. Add \(\frac{1}{4}\) and \(-\frac{1}{4}\) to get \(0\).

Step 4: Repeat Multiply and Add

Repeat the multiply and add step with the new number under the line. Multiply \(0\) by \(-\frac{1}{4}\) to get \(0\), write this under the next coefficient \(-1\), and add to get \(-1\).

Step 5: Continue the Pattern

Continue this pattern: Multiply \(-1\) by \(-\frac{1}{4}\) to get \(\frac{1}{4}\), add this to \(-\frac{1}{4}\) to get \(0\). Multiply \(0\) by \(-\frac{1}{4}\) to get \(0\), add to \(3\) to get \(3\). Multiply \(3\) by \(-\frac{1}{4}\) to get \(-\frac{3}{4}\), add to \(-\frac{5}{4}\) to get \(-2\).

Step 6: Write Down the Result

The numbers below the line are the coefficients of the quotient polynomial, while the last number is the remainder. The quotient is \(x^4 - x^2 + 3\) and the remainder is \(-2\). This yields a final result of \(x^4 - x^2 + 3 - \frac{2}{x + \frac{1}{4}}\).

Key Concepts

Polynomial Division

Polynomial division is a process similar to long division, but it is used with polynomials. Imagine dividing numbers, but instead, you are dividing expressions made up of powers of a variable, often denoted as \(x\). This process involves splitting a polynomial into smaller parts: a quotient and sometimes a remainder. The goal is to divide a dividend by a divisor to find these parts.

The method of synthetic division is a simplified form of polynomial division. It's used primarily when dividing by a linear polynomial of the form \(x - c\). Synthetic division offers convenience because it reduces the complexity and amount of work needed compared to traditional polynomial division. This makes it easier to focus on the coefficients of the polynomial, which significantly streamlines the process.

When using synthetic division:

• Write the coefficients of the polynomial you are dividing.
• Use the zero of the divisor \(x - c\), which is \(c\), for calculations.
• Continue following the pattern of multiplying and adding until you reach the final result.

With these steps, you can efficiently divide a polynomial and gain insights into both the quotient and the remainder.

Remainder Theorem

The remainder theorem plays a crucial role in polynomial division. It states that the remainder of the division of a polynomial \(f(x)\) by a linear divisor \(x - c\) is simply the polynomial evaluated at \(c\), that is \(f(c)\).

This principle not only helps in understanding the outcome of polynomial division but also allows easy calculation of remainders without performing the full division. With synthetic division, the final number at the end of your calculations represents this remainder.

Understanding the remainder theorem can significantly aid you in verifying your work during polynomial divisions. For example:

• After finding the quotient and remainder using synthetic division, you can check if your remainder makes sense by directly evaluating \(f(c)\).
• If all steps are correctly followed, these results should be consistent.

In a given example, if the remainder is \(-2\), it means that \(f(-\frac{1}{4}) = -2\) after division by \(x + \frac{1}{4}\).

Coefficients

Coefficients are the numerical factors in the terms of a polynomial. They are the numbers in front of variable powers, and they determine the shape and position of the polynomial graph.

In synthetic division, coefficients make up the main part of the calculation process. By focusing on just these numbers you save time and effort since there is no need to work with powers of \(x\) explicitly:

• The coefficients of \(x^5 + \frac{1}{4}x^4 - x^3 - \frac{1}{4}x^2 + 3x - \frac{5}{4}\) are \(1, \frac{1}{4}, -1, -\frac{1}{4}, 3, -\frac{5}{4}\).
• These numbers provide a condensed version of the polynomial for easy calculations.
• The synthetic division technique multiplies and adds these coefficients based on the divisor's zero.

These operations leave you with a new set of coefficients representing the quotient polynomial. This transition from one form to another emphasizes the idea that coefficients dictate the behavior of the polynomial.

Polynomial Equations

Polynomial equations are expressions set equal to zero, involving a polynomial set in a variable \(x\). Solving polynomial equations typically means finding a value of \(x\) that makes the equation true.

Synthetic division can help solve polynomial equations by simplifying the polynomial before solving. If a polynomial equation is of higher degree, using division to reduce its degree makes the equation more manageable. It simplifies complexity by dividing the task into smaller pieces.

This process often begins with:

• Using synthetic division to divide polynomial by a linear factor.
• Finding the quotient polynomial as part of simplifying the larger equation.
• This breaks the problem down to either smaller polynomial equations or isolated factors.

Ultimately, the simplified equation can then be solved through factoring or using other algebraic techniques, allowing you to find the roots or solutions of the original polynomial equation.