Problem 14 - Elementary Algebra

Question

\(\left(72 v^{6}-81 v^{4}+54 v^{2}\right) \div\left(9 v^{2}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \( 8v^4 - 9v^2 + 6 \)

Step 1 - Identify the Expression

We start with the expression \( (72v^6 - 81v^4 + 54v^2) / (9v^2) \)

Step 2 - Break Down the Numerator

The numerator is \( 72v^6 - 81v^4 + 54v^2 \). We will divide each term in the numerator by the denominator \( 9v^2 \)

Step 3 - Divide Each Term Separately

Consider each term separately:\( \frac{72v^6}{9v^2} - \frac{81v^4}{9v^2} + \frac{54v^2}{9v^2} \)

Step 4 - Simplify Each Term

Simplify the terms individually:\( \frac{72v^6}{9v^2} = 8v^4 \)\( \frac{81v^4}{9v^2} = 9v^2 \)\( \frac{54v^2}{9v^2} = 6 \)

Step 5 - Combine the Simplified Terms

After simplifying, combine the terms to get:\( 8v^4 - 9v^2 + 6 \)

Key Concepts

Simplifying Polynomials

In algebra, simplifying polynomials involves reducing an expression to its simplest form. The goal is to combine like terms and perform any possible arithmetic to make the expression easier to work with.

For example, take the fraction given in the exercise: \( \left(72 v^{6}-81 v^{4}+54 v^{2}\right) / \left(9 v^{2}\right) \).

To simplify this, we need to break down and reduce each term.
\( How to Simplify Polynomials: \)

• Distribute the divisor to each term individually.
• Divide numerical coefficients by the divisor.
• Reduce the powers of the variables by subtracting the exponent of the divisor from the exponent of each term.

In the exercise, we divided every term by \( 9v^{2} \) and observed how each monomial term simplified, resulting in \( 8v^{4} - 9v^{2} + 6 \).

This method helps us streamline complex polynomials and make them easier to manipulate in future calculations.

Division of Monomials

Understanding the division of monomials is a key concept in algebra. A monomial is a single term polynomial, like \( 72v^{6} \) or \( 9v^{2} \).

When we divide monomials, we split the coefficients and reduce the exponents by performing arithmetic operations separately for numeric and variable parts.
Here’s how it’s done:

• Divide the coefficients: If you have \(72 \div 9\), you get 8.
• Subtract the exponents of like variables: For \( v^{6} / v^{2} \), you subtract the exponent in the divisor from the exponent in the dividend, getting \( v^{4} \).

By applying this method, each term of the given expression \( \left(72 v^{6}-81 v^{4}+54 v^{2}\right) \div \left(9 v^{2}\right) \) is transformed individually as \( 8v^4, -9v^2, \) and \( 6 \).

This systematic approach ensures clarity and simplifies further algebraic operations.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like +, -, *, and /) without an equality sign. Understanding the structure and components of algebraic expressions is crucial for manipulating them effectively.

In the exercise, we start with the algebraic expression: \( \left(72 v^{6}-81 v^{4}+54 v^{2}\right) \div \left(9 v^{2}\right) \).
This expression includes:

• Terms: Individual parts separated by + or - signs.
• Coefficients: Numerical parts of the terms (like 72, -81, and 54).
• Variables: Letters representing numbers (v in this case).
• Exponents: Powers to which the variables are raised (6, 4, and 2).

To solve such expressions, we need to understand the order of operations and how each component interacts.

The exercise leads you through breaking down terms and simplifying them individually, showcasing how algebraic rules make complex expressions manageable and solvable.

Mastering algebraic expressions equips students with the tools needed to navigate higher-level math confidently.