Problem 56 - Elementary Algebra

Question

\(\left(x^{2}+10 x+18\right) \div(x+2)\)

Step-by-Step Solution

Answer

The quotient is \(x + 8 + \frac{2}{x + 2}\).

Step 1: Set Up the Polynomial Long Division

Write the dividend \(\text{x}^{2} + 10\text{x} + 18\) and the divisor \(x + 2\).

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(\text{x}^{2}\) by the leading term of the divisor \(x\), which gives \(x\).

Step 3: Multiply and Subtract

Multiply \x\ by the divisor \(x + 2\), which results in \x^{2} + 2x\. Subtract this from the original dividend to get the new dividend of \(8x + 18\).

Step 4: Repeat the Process

Divide the new leading term \(8x\) by the leading term of the divisor \(x\), giving \8\. Multiply \8\ by \(x + 2\), which results in \8x + 16\. Subtract this result from the new dividend to get the remainder \2\.

Step 5: Write the Final Answer

Combine the quotient parts and add the remainder over the divisor: \(x + 8 + \frac{2}{x + 2}\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another polynomial. It's similar to long division with numbers but applies to algebraic expressions. In our given example, we divide \(\text{x}^2 + 10\text{x} + 18\) by \(x + 2\).

The process involves several steps:

• Set up the division problem by writing the dividend and divisor.
• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by the result obtained in the previous step and subtract this from the dividend.
• Repeat the process with the new dividend obtained.
• Combine the quotient parts and add any remainder over the divisor.

By following these steps, you can break down complex polynomial division into manageable parts.

Division of Polynomials

Division of polynomials can seem challenging, but breaking it down step-by-step makes it more approachable. Let's look into each step in our example dividing \( x^2 + 10x + 18 \) by \ (x + 2) \.

Step-by-Step Breakdown:

• Step 1: Write the division problem with the dividend (\text{x}^2 + 10\text{x} + 18) and the divisor (\text{x} + 2).
• Step 2: Divide the leading term of the dividend (\text{x}^2) by the leading term of the divisor (\text{x}) to get the first term of the quotient (\text{x}).
• Step 3: Multiply the entire divisor (\text{x} + 2) by this quotient term (\text{x}) to get \text{x}^2 + 2\text{x} and subtract this from the original dividend to form a new dividend of 8x + 18.
• Step 4: Repeat the process with the new leading term (8x divided by \text{x} is 8). Multiply \text{x + 2} by 8 to get 8x + 16 and subtract it from the new dividend to get the remainder 2.
• Step 5: Combine quotient parts and write the final answer \text{x + 8 + \frac{2}{x + 2}}.

Algebraic Techniques

Understanding algebraic techniques is crucial for solving polynomial division problems effectively. One key technique is aligning terms properly.

Proper Alignment

• Ensure like terms are aligned in columns (x-terms under x-terms, constant terms under constant terms).
• Use placeholders (0x) for missing degree terms to avoid mistakes.

Multiplying Polynomials

• When multiplying terms, distribute each part of the divisor to each term of the quotient term.
• Carefully perform multiplication and then subtraction, making sure to change the sign of each term you subtract.

Combining Quotient Parts

• As you find each part of the quotient, combine them.
• Any remainder should be written as a fraction where the numerator is the remainder and the denominator is the original divisor.

Elementary Algebra

Elementary algebra lays the foundation for understanding polynomial division. Basic operations such as addition, subtraction, multiplication, and division of algebraic expressions are essential.

Key Concepts in Elementary Algebra:

• Understanding Variables: Variables represent unknown values.
• Order of Operations: Follow the correct order (PEMDAS/BODMAS) when performing calculations.
• Like Terms: Combine like terms to simplify expressions.

Applying Elementary Algebra in Polynomial Division:

• Identify the highest degree term in the dividend and divide it by the highest degree term in the divisor.
• Perform algebraic operations to multiply and subtract terms methodically.
• Repeat the process until you can no longer divide, indicating that you are left with a remainder.

These foundational principles ensure you grasp more sophisticated algebraic techniques used in polynomial division.