Problem 22 - Algebra and Trigonometry

Question

Find the quotient and remainder using long division. \(\frac{2 x^{5}-7 x^{4}-13}{4 x^{2}-6 x+8}\)

Step-by-Step Solution

Answer

The quotient is \(\frac{1}{2}x^3 - x^2 - \frac{5}{2}x + \frac{25}{4}\) and the remainder is \(24x - 13\).

Step 1: Set Up the Long Division

Write the dividend, which is the polynomial \(2x^5 - 7x^4 - 13\), under the long division symbol. Write the divisor, which is \(4x^2 - 6x + 8\), outside the division symbol. Make sure to align the terms by their degree.

Step 2: Divide the First Term

Divide the leading term of the dividend, \(2x^5\), by the leading term of the divisor, \(4x^2\). This gives you \(\frac{2x^5}{4x^2} = \frac{1}{2}x^3\). Write this term above the division bar.

Step 3: Multiply and Subtract

Multiply the entire divisor \(4x^2 - 6x + 8\) by \(\frac{1}{2}x^3\). This gives \(2x^5 - 3x^4 + 4x^3\). Subtract this product from the original dividend \(2x^5 - 7x^4 - 13\) to obtain \(-4x^4 - 4x^3 - 13\).

Step 4: Repeat Division

Take the new dividend \(-4x^4 - 4x^3 - 13\) and divide the leading term \(-4x^4\) by the leading term of the divisor \(4x^2\). This results in \(-x^2\). Write this term above the division bar next to the previous term.

Step 5: Multiply and Subtract Again

Multiply the divisor \(4x^2 - 6x + 8\) by \(-x^2\). The result is \(-4x^4 + 6x^3 - 8x^2\). Subtract this from \(-4x^4 - 4x^3 - 13\), resulting in \(-10x^3 + 8x^2 - 13\).

Step 6: Repeat Same Process

Divide the leading term \(-10x^3\) by \(4x^2\) to get \(-\frac{5}{2}x\). Multiply \(4x^2 - 6x + 8\) by \(-\frac{5}{2}x\) and subtract the result from \(-10x^3 + 8x^2 - 13\).

Step 7: Final Division

Continue dividing with the new polynomial obtained until the degree of the remainder is less than the degree of the divisor. This involves dividing \(25x\) by \(4x^2\) getting \(\frac{25}{4}\), multiply it with divisor and subtract from remainder.

Step 8: Conclusion

The quotient is in the form of \(\frac{1}{2}x^3 - x^2 - \frac{5}{2}x + \frac{25}{4}\), and the remainder is what is left after the last subtraction, which is a polynomial of degree less than the divisor, \(24x - 13\).

Key Concepts

Long Division Method

The long division method is a simple but precise procedure to solve polynomial division problems, similar to how you divide numbers in arithmetic. It helps in finding the quotient and remainder when dividing polynomials. To use the long division method, you need to follow these critical steps:

• Setting Up: Place the dividend inside and the divisor outside the division symbol.
• Division: Begin by dividing the highest degree term of the dividend by the highest degree term of the divisor.
• Multiplication: Multiply the entire divisor by the term obtained in the division and write the result underneath the dividend.
• Subtraction: Subtract this result from your dividend.
• Repeat: Repeat the process with the new polynomial formed after subtraction until the degree of the remainder is less than the degree of the divisor.

By following this methodical process, you'll be able to unravel the components of a polynomial division effectively.

Quotient and Remainder

In polynomial division, the quotient and remainder are two essential terms that make up the division's result. Let's break down these concepts:

• Quotient: This is the result you get after dividing the dividend by the divisor. In the example provided, the quotient is represented as a polynomial \(\frac{1}{2}x^3 - x^2 - \frac{5}{2}x + \frac{25}{4}\).
• Remainder: This is what is left over once the division is complete. It must be of a lesser degree than the divisor. In this case, it is \(24x - 13\).

Together, the quotient and remainder provide a comprehensive picture of how a polynomial divides into another, telling us exactly what the result of the division is without any fractional components. This concept is vital for further mathematical exploration and solving various algebraic equations.

Degree of Polynomial

The degree of a polynomial is a crucial aspect when performing division. It determines when we should stop our long division process. The degree is the highest exponent of the variable within a polynomial. Understanding this concept helps you to:

• Identify When to Stop: Continue dividing until the degree of the remainder is lower than that of the divisor.
• Align Terms Correctly: Keeping terms aligned by degree ensures accuracy in subtraction and further calculations.
• Simplify Calculations: By focusing on the leading terms, the process becomes simpler, and unnecessary steps are avoided.

The degree guides the steps in polynomial division, making sure we perform each step correctly and logically to reach the complete and final result.

Division Algorithm for Polynomials

This algorithm is a fundamental aspect of polynomial division that states any polynomial \(f(x)\) can be expressed as \(d(x)\cdot q(x) + r(x)\) where \(d(x)\) is the divisor, \(q(x)\) is the quotient, and \(r(x)\) is the remainder. This idea articulates the basic structure of polynomial division:

• Formula Formation: The representation using the division algorithm shows how each component fits together.
• No Higher Degree Remainder: Ensures that the remainder's degree is less than that of the divisor, forming a clear end to the division.
• Application in Solving Problems: Helps solve equations and discover roots, intersections, and other algebraic explorations.

The division algorithm not only defines the outcome of polynomial division but also provides a clear path to understanding and resolving polynomial equations. It's a critical tool for mathematicians dealing with algebraic structures and complexities.