Problem 20 - Intermediate Algebra

Question

Perform each division. \(\frac{11 a^{3}-99 a^{2}}{11}\)

Step-by-Step Solution

Answer

The result is \(a^3 - 9a^2\).

Step 1: Simplify Each Term

Start by dividing each term of the polynomial in the numerator by the denominator, 11. The first term is \(\frac{11a^3}{11}\) and the second term is \(\frac{-99a^2}{11}\). This simplifies to \(a^3\) and \(-9a^2\) respectively.

Step 2: Combine Simplified Terms

Combine the simplified terms from Step 1 to write the expression as a polynomial. The division gives us \(a^3 - 9a^2\).

Key Concepts

Simplifying Polynomials

When we're discussing polynomials, we often refer to expressions that involve variables raised to powers and their coefficients. Simplifying polynomials is a key skill you need to master to make working with these algebraic expressions easier. It involves reducing the expressions into their simplest form, making them much easier to handle. A typical simplification process includes:

• Combining like terms, which have the same variable raised to the same power.
• Reducing coefficients through division or factoring common terms.

In our example, we simplified the polynomial \( \frac{11 a^{3}-99 a^{2}}{11} \) by dividing each term by 11. This is a straightforward approach since the denominator is a constant value. This process turns the fraction into a simpler expression, ready for further operations or insights.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators like addition and subtraction. They form the basic building blocks of algebra, enabling us to describe relationships and changes mathematically.An algebraic expression can be as simple as a single term, like \( 3x \), or more complex with multiple terms, like \( a^3 - 9a^2 \). The expression we worked with, \( 11a^3 - 99a^2 \), is a polynomial, which is just a specific type of algebraic expression. Recognizing and categorizing expressions helps determine the most suitable methods for simplification, solving or manipulation. To handle algebraic expressions effectively:

• Identify coefficients and variables.
• Understand the role of each term.
• Apply appropriate mathematical operations to simplify.

Step-by-Step Solutions

Learning mathematics is like building a toolkit of strategies. Step-by-step solutions are one of the most valuable tools. They allow you to break down complex problems into manageable parts. Each step builds on the one before, leading to a final simplified answer.Consider the step-by-step solution for simplifying \( \frac{11 a^{3}-99 a^{2}}{11} \). We started by dividing each term separately:

• First, divide \( 11a^3 \) by 11, simplifying it to \( a^3 \).
• Next, divide \(-99a^2 \) by 11, simplifying it to \(-9a^2 \).

This method not only reduces errors but also enhances understanding and retention. By laying out each step, you see why and how the simplification works, reinforcing knowledge and confidence in handling similar problems in the future. Building this skillset allows for greater mathematical competency.