Problem 15 - Algebra and Trigonometry

Question

Find the quotient and remainder using long division. \(\frac{x^{3}+3 x^{2}+4 x+3}{3 x+6}\)

Step-by-Step Solution

Answer

Quotient: \(\frac{1}{3}x^2 + \frac{1}{3}x + \frac{2}{3}\), Remainder: \(-1\).

Step 1: Set Up the Problem

Write down the long division setup by placing the dividend, \(x^3 + 3x^2 + 4x + 3\), under the division bracket and the divisor, \(3x + 6\), outside the bracket.

Step 2: Divide the First Term

Compare the first term of the dividend, \(x^3\), with the first term of the divisor, \(3x\). Determine what \(x^3\) needs to be multiplied by to get a leading term of \(3x\). We find that \(x^3 \div 3x = \frac{1}{3}x^2\). Write \(\frac{1}{3}x^2\) above the division bar.

Step 3: Multiply and Subtract

Multiply \(\frac{1}{3}x^2\) by \(3x + 6\), giving \(x^3 + 2x^2\). Subtract \(x^3 + 2x^2\) from \(x^3 + 3x^2 + 4x + 3\), which results in \(x^2 + 4x + 3\).

Step 4: Repeat the Division Process

Now, take \(x^2 + 4x + 3\) and repeat the division process. Divide the first term \(x^2\) by \(3x\) to get \(\frac{1}{3}x\). Write this above the division bar.

Step 5: Multiply and Subtract Again

Multiply \(\frac{1}{3}x\) by \(3x + 6\) to get \(x^2 + 2x\). Subtract \(x^2 + 2x\) from \(x^2 + 4x + 3\) to get \(2x + 3\).

Step 6: Final Division Step

Finally, divide the term \(2x\) by \(3x\) to get \(\frac{2}{3}\). Write this above the division bar. Multiply \(\frac{2}{3}\) by \(3x + 6\) to get \(2x + 4\). Subtract \(2x + 4\) from \(2x + 3\), resulting in a remainder of \(-1\).

Step 7: Write the Final Quotient and Remainder

The quotient from the division is \(\frac{1}{3}x^2 + \frac{1}{3}x + \frac{2}{3}\), and the remainder is \(-1\). So, the complete division result is: \[ \frac{x^3 + 3x^2 + 4x + 3}{3x + 6} = \frac{1}{3}x^2 + \frac{1}{3}x + \frac{2}{3} \text{ with a remainder of } -1. \]

Key Concepts

Long Division

Long division in polynomials is an extension of the long division process we learn with numbers. The technique involves dividing a polynomial (the dividend) by another lower-degree polynomial (the divisor) to find the quotient and, sometimes, a remainder. This method is particularly useful when simplifying algebraic expressions. Here’s a simple approach to walk you through the process:

• Set up the division: Write the dividend underneath a long division symbol and the divisor to the left.
• Divide: Focus on the leading term of the dividend and divide it by the leading term of the divisor. Write the result above the division bar.
• Multiply and subtract: Multiply the entire divisor by the term you just wrote and subtract the result from the dividend.
• Repeat: Use the result of the subtraction as a new dividend and repeat the steps until the degree of the remaining polynomial is less than the divisor.

Each step requires careful attention, especially with floating fractions which can appear during division. Practicing these steps makes polynomial division less daunting and more intuitive.

Remainder Theorem

The Remainder Theorem is a useful tool in algebra that helps to determine the remainder of a polynomial division without fully carrying out the division operation. According to this theorem, when a polynomial \(f(x)\) is divided by a linear divisor \(x - c\), the remainder of this division is equal to \(f(c)\). This means you can evaluate the polynomial at \(c\) to find the remainder.This theorem gives us an efficient way to find remainders, allowing us to avoid lengthy calculations. Let’s break it down:

• Identify the divisor in the form \(x - c\).
• Substitute \(c\) into the polynomial.
• The result of this substitution is the remainder.

This theorem is especially helpful in verifying the results of polynomial division and checking work without the need for time-consuming division iterations.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and algebraic operations such as addition, subtraction, multiplication, and division. They are fundamental in mathematics because they allow us to generalize patterns and structures.When dealing with algebraic expressions, especially in polynomial long division, it’s important to understand their forms and how to manipulate them:

• Terms: Individual components such as \(x^2\) or \(3x\).
• Coefficients: Numbers that multiply the variables (\(3\) in \(3x\)).
• Degrees: The highest power of the variable in the expression, such as \(3\) in \(x^3\).

Understanding these elements helps in efficiently carrying out operations like addition, subtraction, or division across polynomials. As you become more familiar with these concepts, working with algebraic expressions becomes more straightforward and less intimidating.