Problem 32 - Intermediate Algebra

Question

Perform each division. \(\frac{6 x^{2}-x-12}{2 x-3}\)

Step-by-Step Solution

Answer

The quotient is \( 3x + 4 \).

Step 1: Arrange for Long Division

To perform polynomial long division, place the polynomial (dividend) \( 6x^2 - x - 12 \) under the division symbol and the divisor \( 2x - 3 \) outside the division symbol.

Step 2: First Division

Divide the first term of the dividend \( 6x^2 \) by the first term of the divisor \( 2x \). This gives \( 3x \). Write \( 3x \) above the division symbol.

Step 3: Multiply and Subtract

Multiply the entire divisor \( 2x - 3 \) by \( 3x \) to get \( 6x^2 - 9x \). Subtract this result from the original polynomial: \[ (6x^2 - x) - (6x^2 - 9x) = 8x \] Bring down the next term from the dividend \( -12 \), making it \( 8x - 12 \).

Step 4: Second Division

Divide the first term of the new dividend \( 8x \) by the first term of the divisor \( 2x \). This gives \( 4 \). Write \( 4 \) above the division symbol after \( 3x \).

Step 5: Multiply and Subtract Again

Multiply the entire divisor \( 2x - 3 \) by \( 4 \) to get \( 8x - 12 \). Subtract this result from the current polynomial \( 8x - 12 \): \[ (8x - 12) - (8x - 12) = 0 \] The remainder is \( 0 \), indicating \( 2x - 3 \) divides the original polynomial entirely.

Key Concepts

Dividing Polynomials

Dividing polynomials is like dividing numbers, but with variables. In polynomial division, you aim to break down a complex polynomial (called the dividend) by using another polynomial (called the divisor). The goal is to express the dividend as the product of the divisor and a quotient, plus a remainder.

In our exercise, we have the dividend, \(6x^2 - x - 12\), and the divisor, \(2x - 3\). The process involves aligning everything within a long division format to make calculations manageable and organized. Start by dividing the leading term of the dividend by the leading term of the divisor, and continue the process of multiplying, subtracting, and bringing down terms until there is no remainder left.

This method ensures all parts of the polynomial are accounted for, preserving the mathematical integrity of your solution.

• Align terms according to their degree.
• Always focus on leading terms for guidance.
• Systematically bring down terms to work on.

Algebraic Division

Algebraic division, particularly polynomial long division, shares similarities with traditional long division, helping us divide expressions where algebra rules the roost. It's practical for simplifying complex expressions and is a foundation for understanding higher mathematics.

This technique involves following a sequence of steps:

• Divide the first term of the polynomial by the first term of the divisor.
• Multiply the entire divisor by the result of the division and subtract from the polynomial.
• Bring down the next term if available.

Using algebraic division, as in our original example, helps to see how each part of the polynomial contributes to the overall outcome. It’s a bit like peeling back layers to see what’s at the core, ensuring every term is properly accounted for through each division cycle.

Remainder Theorem

The Remainder Theorem is a powerful tool in polynomial mathematics, providing insights into the results of division. It states that when a polynomial \(f(x)\) is divided by a linear divisor \(x - a\), the remainder of this division is \(f(a)\).

In the given problem, after performing the polynomial division of \(6x^2 - x - 12\) by \(2x - 3\), the remainder was zero. This outcome signifies that the divisor \(2x - 3\) perfectly divides the dividend with no remainder; essentially, \(2x - 3\) is a factor of the polynomial.

Understanding this theorem helps you check for factors and simplifies polynomial expressions efficiently. And with no remainder, you've successfully confirmed the divisor's role in dismantling the complexity of our original polynomial, ensuring it fits perfectly like a puzzle piece into the subdivision of terms.