Problem 24 - Precalculus Mathematics for Calculus

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

Quotient: \(\frac{1}{2}x^3 - x^2 + \frac{1}{2}x - \frac{5}{4}\); Remainder: \(\frac{-61}{4}x - 13\).

Step 1: Set Up the Long Division

Write down the dividend \(2x^5 - 7x^4 - 13\) under the long division symbol, and the divisor \(4x^2 - 6x + 8\) outside. Arrange the terms of the polynomial in descending order if they are not already.

Step 2: Divide the First Terms

To start the division, divide the first term of the dividend \(2x^5\) by the first term of the divisor \(4x^2\), which results in \(\frac{2x^5}{4x^2} = \frac{1}{2}x^3\). This is the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(\frac{1}{2}x^3\) by the entire divisor \(4x^2 - 6x + 8\), which equals \(2x^5 - 3x^4 + 4x^3\). Subtract this result from the dividend \(2x^5 - 7x^4 - 13\) to get a new polynomial \(-4x^4 - 4x^3 - 13\).

Step 4: Repeat Division Process

Divide the new first term \(-4x^4\) by \(4x^2\) to get \(-x^2\). Multiply \(-x^2\) by the divisor \(4x^2 - 6x + 8\), resulting in \(-4x^4 + 6x^3 - 8x^2\). Subtract this product from the current polynomial \(-4x^4 - 4x^3 - 13\) to obtain \(2x^3 - 8x^2 - 13\).

Step 5: Continue Division Process

Take the resulting polynomial \(2x^3 - 8x^2 - 13\) and divide \(2x^3\) by \(4x^2\) to get \(\frac{1}{2}x\). Multiply this by the divisor and subtract again to get a new polynomial \(-5x^2 - 13\).

Step 6: Finalize Quotient and Remainder

Finally, divide \(-5x^2\) by \(4x^2\), resulting in \(-\frac{5}{4}\) and reverse the multiplication/subtraction process to find the remainder \(\frac{-61}{4}x - 13\). The division is complete when the degree of the remainder is less than the degree of the divisor. Thus, the quotient is \(\frac{1}{2}x^3 - x^2 + \frac{1}{2}x - \frac{5}{4}\) and the remainder is \(\frac{-61}{4}x - 13\).

Key Concepts

Quotient and Remainder

In polynomial long division, similar to dividing numbers, we want to find how many times one polynomial fits into another while taking note of what’s left over. Here, the "quotient" is the result of the division, representing how many complete times the divisor fits into the dividend. The "remainder" is what cannot be completely divided by the divisor and is left over after all possible division steps are completed.

For example, when dividing \(2x^5 - 7x^4 - 13\) by \(4x^2 - 6x + 8\), we find:

• The quotient as \(\frac{1}{2}x^3 - x^2 + \frac{1}{2}x - \frac{5}{4}\)
• The remainder as \(\frac{-61}{4}x - 13\)

This method indicates the quotient is a polynomial that represents the main result of division, while the remainder is another polynomial of a lesser degree than the divisor, showing what’s left.

Polynomial Division Steps

To successfully perform polynomial long division, it's crucial to follow these structured steps that help ensure a correct solution. Here’s a breakdown of the typical technique:

• **Set Up the Division:**
  Arrange the dividend and divisor polynomial such that their terms are in descending order of their degrees.
• **Divide the Leading Terms:**
  Begin by dividing the first term of the dividend by the first term of the divisor. This will be the first term in the quotient.
• **Multiply and Subtract:**
  Multiply the entire divisor by the term you just found and subtract this from the dividend polynomial. This will give a new polynomial.
• **Repeat the Process:**
  Use the new polynomial obtained from subtraction as the following dividend and repeat the process of division until the degree of your new polynomial becomes less than that of the divisor.
• **Find the Remainder:**
  The remaining terms at this point form the remainder, which would have a lesser degree than the divisor polynomial.

Following these steps ensures that every component of the polynomial division is meticulously executed.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition and multiplication. In the context of polynomial division, both the dividend and divisor are algebraic expressions involving variables like \(x\), constants, and different mathematical operations.

Understanding the structure of these expressions is pivotal when performing operations like addition, subtraction, or division with them. For instance, in the expression \(2x^5 - 7x^4 - 13\), \(2x^5\) represents a term where 2 is a coefficient, \(x\) is a variable, and \(5\) is the degree of the term.

Each term in a polynomial must be handled carefully during division to maintain the integrity of its operations. In the division process, recognizing similar degrees of terms between divisor and dividend helps in carrying out consistent multiplications and subtractions effectively.