Problem 12 - Elementary Algebra

Question

\(\left(54 z^{4}-36 z^{3}-6 z^{2}+12 z\right) \div(6 z)\)

Step-by-Step Solution

Answer

9z^3 - 6z^2 - z + 2

Step 1: Divide Each Term Inside the Parentheses by the Divisor

Divide each term in the expression \ \ \(54z^4 - 36z^3 - 6z^2 + 12z\) by 6z separately. This means computing: \ \ - \(\frac{54z^4}{6z}\), \(\frac{-36z^3}{6z}\), \(\frac{-6z^2}{6z}\), and \(\frac{12z}{6z}\).

Step 2: Simplify Each Fraction

Perform the division for each individual term: \ - \(54z^4 \div 6z = 9z^3\) \ - \(36z^3 \div 6z = 6z^2\) \ - \(6z^2 \div 6z = z\) \ - \(12z \div 6z = 2\)

Step 3: Write the Final Simplified Expression

Combine the simplified terms from the previous step to get the final expression: \ \ \(9z^3 - 6z^2 - z + 2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of variables, constants, and operators such as addition, subtraction, multiplication, and division. In our problem, the expression given is \(54z^4 - 36z^3 - 6z^2 + 12z\). Each part of the expression separated by a plus or minus sign is called a 'term.' For example, \(54z^4\) and \(-36z^3\) are terms in the given expression. Understanding how to manipulate these expressions is key to solving algebraic problems. When working with algebraic expressions, always remember to follow the order of operations (PEMDAS/BODMAS rules). This ensures that you simplify the expression correctly. For instance, multiplication and division should be performed before addition and subtraction.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. In our exercise, each term of the polynomial is divided by the given divisor, \(6z\). For instance, \(\frac{54z^4}{6z}\) simplifies to \(9z^3\). To achieve this:

• Divide the coefficients (numbers in front of the variables) as you normally would.
• Subtract the exponent of the divisor's variable (if it exists) from the exponent of the term's variable. This is applying the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\).

Simplifying each term this way ensures that you handle both the numerical and variable parts of fractions correctly. Always rewrite the term ensuring it’s in its simplest form before combining it with other terms in the expression.

Polynomial Simplification

Polynomial simplification involves reducing the given polynomial to its most basic form. In our problem, each term resulted from the division of the polynomial. For instance, after reducing \(\frac{54z^4}{6z}\) to \(9z^3\), our task is to handle each term similarly. After simplifying all terms, you must combine them to form the final simplified polynomial expression. Our result was \(9z^3 - 6z^2 - z + 2\). When simplifying polynomials, it's crucial to:

• Ensure each component is fully simplified.
• Combine like terms (terms with the same base and exponent).

This makes the expression more manageable and easier to understand.

Long Division in Algebra

Long division in algebra works similarly to numerical long division but with added complexity due to variables. In our exercise, we simplified this by breaking down the expression into individual terms for division. The steps include:

• Dividing each term individually by the divisor.
• Reducing each term to its simplest form (as explained in 'Simplifying Fractions').

This method helps systematically reduce complex expressions. The key advantage of long division in algebra is that it simplifies the overall problem into smaller, more manageable parts, ultimately providing the final simplified expression. Always ensure you combine the simplified terms correctly at the end to complete the process.