Problem 17 - College Algebra with Modeling and Visualization

Question

Divide Check your answer. $$\frac{4 x^{3}-x^{2}-5 x+6}{x-1}$$

Step-by-Step Solution

Answer

The quotient is \(4x^2 + 3x - 2\) with a remainder of \(4\).

Step 1: Setup Polynomial Long Division

To divide \( \frac{4x^3 - x^2 - 5x + 6}{x-1} \), write it as a long division problem. The dividend is \( 4x^3 - x^2 - 5x + 6 \) and the divisor is \( x - 1 \). Align them similar to numerical division.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend, \(4x^3\), by the leading term of the divisor, \(x\). This gives \(4x^2\). Write \(4x^2\) above the dividend.

Step 3: Multiply and Subtract

Multiply \(4x^2\) by \(x - 1\) to get \(4x^3 - 4x^2\). Subtract this from the current dividend, \(4x^3 - x^2 - 5x + 6\), which gives \(3x^2 - 5x + 6\).

Step 4: Repeat the Division Process

Divide \(3x^2\) by \(x\), yielding \(3x\). Write \(3x\) next to \(4x^2\) above the dividend. Multiply \(3x\) by \(x - 1\) giving \(3x^2 - 3x\), then subtract from \(3x^2 - 5x + 6\), resulting in \(-2x + 6\).

Step 5: Final Division Step

Divide \(-2x\) by \(x\) to get \(-2\). Write \(-2\) in the answer area. Multiply \(-2\) by \(x - 1\) to obtain \(-2x + 2\), and subtract from \(-2x + 6\) to find the remainder of \(4\).

Step 6: Write the Result

The result from the division is \(4x^2 + 3x - 2\) with a remainder of \(4\). This can be expressed as \(4x^2 + 3x - 2 + \frac{4}{x-1}\).

Step 7: Verify the Solution

To verify, multiply \(4x^2 + 3x - 2 + \frac{4}{x-1}\) by \(x-1\). Simplify to see if you get the original polynomial \(4x^3 - x^2 - 5x + 6\), confirming the division is correct.

Key Concepts

Polynomial Division

Polynomial division, much like regular division, is a method for dividing a polynomial by another polynomial. In this type of division, one polynomial is the dividend (the number being divided), and the other is the divisor. The goal is to split the dividend into the divisor's multiples plus any remainder, if applicable.

There are two main types of polynomial division:

• Long Division: This method is best used when the divisor has more than one term. It follows a step-by-step procedure similar to long division with numbers.
• Synthetic Division: A shorter, simpler method applicable when dividing by linear polynomials (of the form \(x - c\)).

In this exercise, we use long division to divide \(4x^3 - x^2 - 5x + 6\) by \(x-1\). The process involves dividing the leading terms, multiplying to form a product, then subtracting, and repeating the process with the resulting polynomial. Each step gradually reduces the degree of the polynomial until a final result and remainder are achieved.

Algebra

Algebra is the branch of mathematics concerning the study of operations and the rules for manipulating those operations. It often involves symbols and letters representing numbers, called variables, which makes it especially powerful for generalizing mathematical concepts and solving equations.

In polynomial division, algebraic skills are crucial. Here's a breakdown of key algebraic principles used:

• Variables and Constants: Understanding how both interact and create terms in a polynomial (e.g., \(4x^3, -x^2\)).
• Operations on Polynomials: Adding, subtracting, and multiplying polynomials during the division process to manage terms effectively.
• Simplification: Combining like terms (terms with the same variable and exponent) to simplify polynomial expressions during division.

Polynomials provide a way to create and manipulate expressions that model various real-world quantities mathematically, making algebra a fundamental component of learning to divide polynomials correctly.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding them extends a student’s ability to work with polynomials in various forms. In the provided exercise, the final result of the division is expressed as a rational expression: \( 4x^2 + 3x - 2 + \frac{4}{x-1} \).

Key points about rational expressions include:

• Simplification: Much like numerical fractions, rational expressions can often be simplified by canceling common factors.
• Conditions: The division by zero is undefined, so it's essential to note where a denominator might be zero, which in this case, \(x-1\) cannot be zero.
• Usage: Rational expressions are essential in calculus and advanced mathematics since many functions can be expressed as such.

Rational expressions give us powerful tools to work with ratios of polynomials and understanding them is essential for success in more advanced algebra topics.