Problem 18 - Algebra 1

Question

Divide. Divide \(9 m^{3}+4 m^{2}-8 m\) by \(m\).

Step-by-Step Solution

Answer

The simplified polynomial after division is \(9m^{2} + 4m - 8\).

Step 1: Divide each term by \(m\)

We begin by dividing each term of the polynomial by \(m\). This operation is performed as follows:\n\n \(\frac{9m^{3}}{m} + \frac{4m^{2}}{m} - \frac{8m}{m}\)

Step 2: Simplify each term using the properties of exponents

To simplify each term, we use the quotient of power rule, which requires subtracting the exponent of the divisor from the exponent of the dividend. This is done as follows:\n\n \(9m^{3-1} + 4m^{2-1} - 8m^{1-1}\)

Step 3: Perform the subtraction in the exponents and simplify

We now subtract the exponents, and simplify each term to obtain the final result as follows:\n\n \(9m^{2} + 4m - 8\)

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when working with polynomial division and simplifying algebraic expressions. Exponents indicate how many times a number, called the base, is multiplied by itself. For example, in the term \(m^3\), \(m\) is the base and \(3\) is the exponent, indicating that \(m\) is multiplied by itself three times.

When dividing polynomials, you often deal with exponents. Properties of exponents help us simplify the process. The key properties include the product of powers, the quotient of powers, and the power of a power rules.

In particular, the quotient of powers rule, such as \( \frac{a^m}{a^n} = a^{m-n} \), is essential when simplifying terms in polynomial division. This rule helps in reducing the complexity of each term when one polynomial is divided by another with a common base.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with and understand. When dividing polynomials, each term of the polynomial is divided individually, as shown in the problem.

To simplify \( \frac{9m^3}{m} + \frac{4m^2}{m} - \frac{8m}{m} \), each term is divided by \(m\), simplifying the expression considerably. This step is often referred to as 'term-by-term division' and is essential in simplifying complex algebraic expressions.

Once you apply the quotient of powers rule by subtracting the exponents of like bases, each term becomes much simpler. This simplification can be done step by step to maintain clarity in calculations, ensuring accurate results without overlooking any operations.

Quotient of Power Rule

The quotient of power rule is a specific property of exponents used when simplifying expressions involving division of terms with the same base. According to this rule, you subtract the exponent of the denominator from the exponent of the numerator, keeping the base unchanged.

For example, simplifying \( \frac{9m^3}{m} \) using the quotient of power rule, you perform the subtraction \(3 - 1\) in the exponent, resulting in \(9m^{2}\). This step-by-step handling ensures each part of the expression is correctly minimized to its simplest form.

Using the quotient of power rule is essential in algebra, especially for polynomial division, as it makes expressions more manageable and easier to solve. Recognizing and applying this rule allows students to clear complex exponents quickly, facilitating the entire simplification and division process.