Problem 9 - Algebra A Combined Function

Question

Perform each division. See Examples 1 through 3. $$ \frac{-9 x^{5}+3 x^{4}-12}{3 x^{3}} $$

Step-by-Step Solution

Answer

The result is \(-3x^2 + x - \frac{4}{x^3}\).

Step 1: Set up the division

We start by writing the given expression as a single fraction: \[ \frac{-9x^{5} + 3x^{4} - 12}{3x^{3}} \] Our task is to divide each term in the numerator by the monomial in the denominator.

Step 2: Divide each term individually

Divide each term in the numerator by the term in the denominator:1. Divide \(-9x^5\) by \(3x^3\): - Coefficient: \(-9 \div 3 = -3\) - Power of \(x\): \(x^{5-3} = x^2\) Result: \(-3x^2\).2. Divide \(3x^4\) by \(3x^3\): - Coefficient: \(3 \div 3 = 1\) - Power of \(x\): \(x^{4-3} = x^1\) Result: \(x\).3. Divide \(-12\) by \(3x^3\): - Coefficient: \(-12 \div 3 = -4\) - Since there is no \(x\) term in the numerator, write as \(-\frac{4}{x^3}\).Thus, the division yields: \(-3x^2 + x - \frac{4}{x^3}\).

Step 3: Combine results

Write down the results from each division together to form the final simplified expression:\[ -3x^2 + x - \frac{4}{x^3} \]

Key Concepts

Step-by-Step Solution

When solving polynomial division problems, having a clear step-by-step approach can make the process much easier and less intimidating. By breaking down each part of the solution, you will see exactly how to handle each component of the problem, and ultimately, you'll gain a better understanding of the division process in algebra.For the problem \( \frac{-9x^{5} + 3x^{4} - 12}{3x^{3}} \), the first step in our solution is to set up the division. This involves writing the expression as a single fraction. The numerator is \(-9x^{5} + 3x^{4} - 12\) and the denominator is \(3x^{3}\). This setup is crucial as it allows us to clearly see what is being divided.Next, we handle each term separately by dividing the terms in the numerator by the monomial in the denominator. This step-by-step breakdown involves:

• Dividing \(-9x^5\) by \(3x^3\) results in \(-3x^2\).
• Dividing \(3x^4\) by \(3x^3\) gives us \(x\).
• Finally, dividing \(-12\) by \(3x^3\) results in \(-\frac{4}{x^3}\).

Once these calculations are complete, combining the results provides the final answer of \(-3x^2 + x - \frac{4}{x^3}\). Understanding this systematic approach will greatly enhance your ability to tackle similar division problems with confidence.

Monomial Division

Monomial division is a fundamental concept in algebra where you divide each term of a polynomial by a monomial. A monomial is a single term expression like \(3x^3\) from our problem.The process involves dividing the coefficients and the variable parts separately. For instance, when dividing \(-9x^{5}\) by \(3x^{3}\), you first divide the coefficients: \(-9 \div 3 = -3\). Then, handle the powers of \(x\) by subtracting the exponents: \(5-3=2\). This gives the result, \(-3x^2\).This approach is repeated for each term in the polynomial, ensuring that both any numbers and variables are properly divided. It's important to remember:

• Subtract the exponents of like bases.
• If a term in the numerator does not have a variable that matches the denominator, divide the constants, and keep any remaining powers of variables in the denominator, as with \(-\frac{4}{x^3}\).

By mastering monomial division, you simplify and solve algebraic expressions with ease, making complex expressions much more manageable.

Simplifying Expressions

Simplifying expressions is a critical skill in algebra that makes equations easier to work with. It involves reducing expressions to their simplest form.In our division problem, simplifying the result follows the process of combining like terms and ensuring the expression is as straightforward as possible. Each term in the quotient \(-3x^2 + x - \frac{4}{x^3}\) is already simplified, as there are no like terms to combine and each fraction is in its reduced form.The primary goals of simplification are to:

• Make the expression easier to understand and solve.
• Minimize the complexity by reducing fractions and combining similar parts.
• Present the solution in the cleanest possible way, which is critical for subsequent algebraic operations.

Learning to simplify effectively allows you to solve and manipulate algebraic equations with improved accuracy and efficiency. It sets the stage for solving more advanced mathematical problems confidently.