Problem 24 - College Algebra with Modeling and Visualization

Question

Divide the expression. $$\frac{20 x^{4}+6 x^{3}-2 x^{2}+15 x-2}{5 x-1}$$

Step-by-Step Solution

Answer

The quotient is \(4x^3 + 2x^2 + 3\) with a remainder of \(1\).

Step 1: Set Up the Division

To begin dividing, we need to set up the polynomial division. We will divide the polynomial \(20x^4 + 6x^3 - 2x^2 + 15x - 2\) by \(5x - 1\) using long division.

Step 2: Divide the Leading Terms

Take the leading term of the dividend \(20x^4\) and divide it by the leading term of the divisor \(5x\). This gives us \(4x^3\). Write \(4x^3\) above the division line.

Step 3: Multiply and Subtract

Multiply \(4x^3\) by \(5x - 1\), which results in \((4x^3)(5x) - (4x^3)(1) = 20x^4 - 4x^3\). Subtract this from the original polynomial to get \(0x^4 + (6x^3 + 4x^3) - 2x^2 + 15x - 2 = 10x^3 - 2x^2 + 15x - 2\).

Step 4: Repeat the Process

Divide the new leading term \(10x^3\) by \(5x\) to get \(2x^2\). Multiply \(2x^2\) by \(5x - 1\) to get \(10x^3 - 2x^2\). Subtract \(10x^3 - 2x^2\) from \(10x^3 - 2x^2 + 15x - 2\) to obtain \(0 + 15x - 2 = 15x - 2\).

Step 5: Continue Division

Divide the next leading term \(15x\) by \(5x\) to get \(3\). Multiply \(3\) by \(5x - 1\) to get \(15x - 3\). Subtract \(15x - 3\) from \(15x - 2\) to find that the remainder is \(1\).

Step 6: Write the Final Answer

The quotient is \(4x^3 + 2x^2 + 3\) and the remainder is \(1\). So, the division result is \(4x^3 + 2x^2 + 3 + \frac{1}{5x - 1}\).

Key Concepts

Long Division

Long division isn't just for numbers; it can be applied to polynomials as well! It’s a methodical process used to divide a polynomial by another polynomial. The idea is similar to numerical long division, where you break down the process step by step.

• Identify the leading term of both the dividend (the polynomial you're dividing) and the divisor.
• Divide the leading term of the dividend by the leading term of the divisor. This gives you the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the original dividend.
• Repeat these steps with the new dividend obtained after subtraction.

The process continues until you reach a remainder that is smaller than the degree of the divisor.

For example, when dividing the polynomial \(20x^4 + 6x^3 - 2x^2 + 15x - 2\) by \(5x - 1\), you proceed iteratively, obtaining terms like \(4x^3, 2x^2, \) and \(3\), until only the remainder is left. This method provides not only the quotient but also helps us understand the remainder in polynomial division.

Remainder Theorem

The Remainder Theorem is a nifty concept that connects polynomial division to finding remainders quickly. If a polynomial \(f(x)\) is divided by a linear divisor \(x - a\), the remainder of this division is simply \(f(a)\).

This implies that, instead of going through the whole division process, you can plug \(a\) into the polynomial and quickly find out what the remainder is!

• First, identify the value of \(a\) in your divisor \(x-a\).
• Substitute \(a\) into the polynomial \(f(x)\).
• The resulting value is your remainder.

For example, with the division problem given earlier, the divisor is \(5x - 1\), which translates to \(x - \frac{1}{5}\). Substituting \(\frac{1}{5}\) into the polynomial yields the remainder.

Use the Remainder Theorem to verify or shortcut some steps in polynomial division. It's a quick check and a great way to prevent mistakes.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Like any fraction, you can perform operations such as addition, subtraction, multiplication, and division.

When dividing rational expressions, you're essentially dividing polynomials, often employing either long division or synthetic division when the divisor is simple.

• Simplify where possible by canceling common factors.
• Ensure the divisor is not zero by checking the expression’s domain.
• Rewrite the division as multiplication of the reciprocal when necessary.

In the exercise, dividing the expression \(\frac{20x^4 + 6x^3 - 2x^2 + 15x - 2}{5x - 1}\) results in another rational expression with a polynomial quotient and a remainder term often expressed as a fraction.

Understanding rational expressions helps with more advanced topics in algebra and calculus, where such divisions regularly appear in problems involving complex fractions and solutions to equations.