Problem 37 - Intermediate Algebra

Question

Perform each division. \(\frac{6 x^{3}+11 x^{2}-19 x-2}{3 x-2}\)

Step-by-Step Solution

Answer

The quotient is \( 2x^2 + 5x - 3 \) with a remainder of \( -8 \), expressed as \( 2x^2 + 5x - 3 + \frac{-8}{3x - 2} \).

Step 1: Set Up the Division

To perform the polynomial division \( \frac{6x^3 + 11x^2 - 19x - 2}{3x - 2} \), set it up like a long division problem. Here, \( 6x^3 + 11x^2 - 19x - 2 \) is the dividend and \( 3x - 2 \) is the divisor.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \( 6x^3 \) by the leading term of the divisor \( 3x \): \( \frac{6x^3}{3x} = 2x^2 \). This is the first part of the quotient.

Step 3: Multiply and Subtract

Multiply \( 2x^2 \) by the entire divisor \( 3x - 2 \), which gives \( 6x^3 - 4x^2 \). Subtract \( 6x^3 - 4x^2 \) from \( 6x^3 + 11x^2 \), resulting in \( 15x^2 \).

Step 4: Bring Down the Next Term

Bring down the next term from the dividend, \( -19x \), to form the new dividend, \( 15x^2 - 19x \).

Step 5: Divide Again

Divide the new leading term \( 15x^2 \) by \( 3x \): \( \frac{15x^2}{3x} = 5x \). Add this to the quotient.

Step 6: Multiply and Subtract Again

Multiply \( 5x \) by \( 3x - 2 \), obtaining \( 15x^2 - 10x \). Subtract \( 15x^2 - 10x \) from \( 15x^2 - 19x \), which results in \( -9x \).

Step 7: Bring Down the Final Term

Bring down the last term from the original dividend, \( -2 \), giving \( -9x - 2 \) as the new dividend.

Step 8: Final Division

Divide \( -9x \) by \( 3x \): \( \frac{-9x}{3x} = -3 \). Add \( -3 \) to the quotient.

Step 9: Multiply and Subtract a Final Time

Multiply \( -3 \) by \( 3x - 2 \), resulting in \( -9x + 6 \). Subtract this from \( -9x - 2 \) to find the remainder: \( -8 \).

Step 10: Write the Final Answer

The division is complete. The quotient is \( 2x^2 + 5x - 3 \) with a remainder of \( -8 \). So, the division results in \( 2x^2 + 5x - 3 + \frac{-8}{3x - 2} \).

Key Concepts

Long Division

Long Division is a method used to divide larger numbers or polynomials into smaller, manageable parts. It’s similar to the division you learned early in school, but instead of numbers, you now work with polynomials, which are expressions that contain variables. When it comes to polynomial division, you divide the polynomial (called the dividend) by another polynomial (the divisor) following a series of steps. In our example, we perform long division on

• The dividend: \(6x^3 + 11x^2 - 19x - 2\)
• The divisor: \(3x - 2\)

The process begins by arranging these polynomials into a division setup.
You then perform a series of quotient and subtraction operations that progressively reduce the scale of the problem.
This structured approach makes sure that every step logically builds upon the previous one, gradually bringing you closer to a complete solution.

Quotient and Remainder

The concepts of quotient and remainder are crucial to understanding polynomial division through long division. When dividing, the quotient represents the result you get outside the division symbol, while the remainder is the portion that couldn't be evenly divided. In our specific polynomial division of

• Dividend: \(6x^3 + 11x^2 - 19x - 2\)
• Divisor: \(3x - 2\)

after carrying out all the necessary steps, the quotient is determined to be \(2x^2 + 5x - 3\) and the remainder is \(-8\).
This means the original polynomial can be expressed as the quotient plus the remainder divided by the divisor: \[ 2x^2 + 5x - 3 + \frac{-8}{3x - 2} \] It's akin to figuring out how many times a divisor can "fit" into a dividend, leaving a remainder that is too small to be divided any further by the divisor.

Leading Terms

Leading terms are the highest degree terms in polynomials that provide guidance throughout the process of polynomial division. When we perform the division, we first look at dividing the leading term of the dividend by the leading term of the divisor. This provides an initial quotient that kicks off our division process. In our case:

• The leading term of the dividend is \(6x^3\)
• The leading term of the divisor is \(3x\)

Our first step is to divide \(6x^3\) by \(3x\), resulting in \(2x^2\).
This term \(2x^2\) is then used to multiply the entire divisor \(3x - 2\), and the result is subtracted from the dividend or intermediate results.
By repeating this process, dealing primarily with leading terms first, the polynomial degree is gradually reduced, and the division proceeds smoothly until the completion or until the degree of the remainder is less than that of the divisor. This technique simplifies complex polynomial problems by focusing on the highest power terms first.